「数学」の記事 - Crieit

素因数分解を思い出そうぜ（＾～＾）？

2023-11-29T20:54:08+09:00

きふわらべRPG で素因数分解を使う

「　きふわらべRPGで　素因数分解したいんで、素因数分解を思い出そうぜ？」

「　１人でやってくれだぜ」

「　合成数を　素数だけで　割り切ればいいのよ」

「　👇　割るしかないの？」

📖　素因数分解とは？やり方を5つのステップでわかりやすく解説【例題・応用問題付き】

「　👇　ポラードのロー法 というのがあるらしいぜ」

📖　素因数分解の高速なアルゴリズム（ロー法）

「　分からん。割ることにする」

ＲＰＧの戦闘シーンのダメージ計算で使う

ある正の整数 n があるとする。
とりあえず n は 1～100 ぐらいを考えておく。

n を　2, 3, 5, 7　の素数で素因数分解する。

答えがいくつかあるとき、大きな数を多く使ったものを　１つ　選ぶことにする。

「　👆　じゃあ　試しに　n=30　ならどうだぜ？」

「　５×５かける……、５×３×２＝３０、　５×３×２　しかなくない？」

「　どういうアルゴリズムだぜ？」

「　だから　割るのよ」

愚直に書くと

# 素因数分解の練習

print("""\
ある正の整数 n があるとする。
とりあえず n は 1～100 ぐらいを考えておく。

n を　2, 3, 5, 7　の素数で素因数分解する。

答えがいくつかあるとき、大きな数を多く使ったものを　１つ　選ぶことにする。
""")

print("Please input number:")
n = int(input())

def factorize(n):

    for a in reversed(range(1,101)):
        aa = 7 ** a

        # n 以下の小さな合成数 7a なら、とりあえず n から、それを引く
        if aa <= n:
            remain = n - aa
            print(f"(7x{a}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [a, 0, 0, 0]

            # 余った数で続きをやる

            for b in reversed(range(1,101)):
                bb = aa * 5 ** b

                if bb <= n:
                    remain = n - bb
                    print(f"\t(7x{a} x 5x{b}) remain:{remain}")

                    # 割り切れた
                    if remain == 0:
                        return [a, b, 0, 0]

                    # 余った数で続きをやる

                    for c in reversed(range(1,101)):
                        cc = bb * 3 ** c

                        if cc <= n:
                            remain = n - cc
                            print(f"\t\t(7x{a} x 5x{b} x 3x{c}) remain:{remain}")

                            # 割り切れた
                            if remain == 0:
                                return [a, b, c, 0]

                            # 余った数で続きをやる

                            for d in reversed(range(1,101)):
                                dd = cc * 2 ** d

                                if dd <= n:
                                    remain = n - dd
                                    print(f"\t\t\t(7x{a} x 5x{b} x 3x{c} x 2x{d}) remain:{remain}")

                                    # 割り切れた
                                    if remain == 0:
                                        return [a, b, c, d]

    for b in reversed(range(1,101)):
        bb = 5 ** b

        if bb <= n:
            remain = n - bb
            print(f"(5x{b}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [0, b, 0, 0]

            # 余った数で続きをやる

            for c in reversed(range(1,101)):
                cc = bb * 3 ** c

                if cc <= n:
                    remain = n - cc
                    print(f"\t(5x{b} x 3x{c}) remain:{remain}")

                    # 割り切れた
                    if remain == 0:
                        return [0, b, c, 0]

                    # 余った数で続きをやる

                    for d in reversed(range(1,101)):
                        dd = cc * 2 ** d

                        if dd <= n:
                            remain = n - dd
                            print(f"\t\t(5x{b} x 3x{c} x 2x{d}) remain:{remain}")

                            # 割り切れた
                            if remain == 0:
                                return [0, b, c, d]

    for c in reversed(range(1,101)):
        cc = 3 ** c

        if cc <= n:
            remain = n - cc
            print(f"(3x{c}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [0, 0, c, 0]

            # 余った数で続きをやる

            for d in reversed(range(1,101)):
                dd = cc * 2 ** d

                if dd <= n:
                    remain = n - dd
                    print(f"\t(3x{c} x 2x{d}) remain:{remain}")

                    # 割り切れた
                    if remain == 0:
                        return [0, 0, c, d]

    for d in reversed(range(1,101)):
        dd = 2 ** d

        if dd <= n:
            remain = n - dd
            print(f"(2x{d}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [0, 0, 0, d]

print(f"Anser:{factorize(n)}")

「　👆　愚直に書くと　こうか」

「　ややこしいから　一般化しなくてよくない？」

n = 1～100 まで見てみよう

「　👇　n を 1 から 100 まで入れてみようぜ？」

N=1 Answer=None
N=2 Answer=[0, 0, 0, 1]
N=3 Answer=[0, 0, 1, 0]
N=4 Answer=[0, 0, 0, 2]
N=5 Answer=[0, 1, 0, 0]
N=6 Answer=[0, 0, 1, 1]
N=7 Answer=[1, 0, 0, 0]
N=8 Answer=[0, 0, 0, 3]
N=9 Answer=[0, 0, 2, 0]
N=10 Answer=None
N=11 Answer=None
N=12 Answer=[0, 0, 1, 2]
N=13 Answer=None
N=14 Answer=None
N=15 Answer=[0, 1, 1, 0]
N=16 Answer=[0, 0, 0, 4]
N=17 Answer=None
N=18 Answer=[0, 0, 2, 1]
N=19 Answer=None
N=20 Answer=None
N=21 Answer=None
N=22 Answer=None
N=23 Answer=None
N=24 Answer=[0, 0, 1, 3]
N=25 Answer=[0, 2, 0, 0]
N=26 Answer=None
N=27 Answer=[0, 0, 3, 0]
N=28 Answer=None
N=29 Answer=None
N=30 Answer=[0, 1, 1, 1]
N=31 Answer=None
N=32 Answer=[0, 0, 0, 5]
N=33 Answer=None
N=34 Answer=None
N=35 Answer=[1, 1, 0, 0]
N=36 Answer=[0, 0, 2, 2]
N=37 Answer=None
N=38 Answer=None
N=39 Answer=None
N=40 Answer=None
N=41 Answer=None
N=42 Answer=None
N=43 Answer=None
N=44 Answer=None
N=45 Answer=[0, 1, 2, 0]
N=46 Answer=None
N=47 Answer=None
N=48 Answer=[0, 0, 1, 4]
N=49 Answer=[2, 0, 0, 0]
N=50 Answer=None
N=51 Answer=None
N=52 Answer=None
N=53 Answer=None
N=54 Answer=[0, 0, 3, 1]
N=55 Answer=None
N=56 Answer=None
N=57 Answer=None
N=58 Answer=None
N=59 Answer=None
N=60 Answer=[0, 1, 1, 2]
N=61 Answer=None
N=62 Answer=None
N=63 Answer=None
N=64 Answer=[0, 0, 0, 6]
N=65 Answer=None
N=66 Answer=None
N=67 Answer=None
N=68 Answer=None
N=69 Answer=None
N=70 Answer=None
N=71 Answer=None
N=72 Answer=[0, 0, 2, 3]
N=73 Answer=None
N=74 Answer=None
N=75 Answer=[0, 2, 1, 0]
N=76 Answer=None
N=77 Answer=None
N=78 Answer=None
N=79 Answer=None
N=80 Answer=None
N=81 Answer=[0, 0, 4, 0]
N=82 Answer=None
N=83 Answer=None
N=84 Answer=None
N=85 Answer=None
N=86 Answer=None
N=87 Answer=None
N=88 Answer=None
N=89 Answer=None
N=90 Answer=[0, 1, 2, 1]
N=91 Answer=None
N=92 Answer=None
N=93 Answer=None
N=94 Answer=None
N=95 Answer=None
N=96 Answer=[0, 0, 1, 5]
N=97 Answer=None
N=98 Answer=None
N=99 Answer=None
N=100 Answer=None

「　👆　20 は 5*2*2 だろ。ミスってんのかな？」

「　お父んのループ、３の倍数が０のケースを　表現できてないんだぜ」

「　なるほど」

0乗は1

「　0乗は1 だから、ループで 0 まで回せばいいんじゃないの？」

「　それだぜ」

# 素因数分解の練習
#
# cd math/prime_factorization
# python step1.py

def factorize(n, debug=True):

    for a in reversed(range(0,101)):
        aa = 7 ** a

        # n 以下の小さな合成数 7a なら、とりあえず n から、それを引く
        if aa <= n:
            remain = n - aa

            if debug:
                print(f"(7x{a}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [a, 0, 0, 0]

            # 余った数で続きをやる

            for b in reversed(range(0,101)):
                bb = aa * 5 ** b

                if bb <= n:
                    remain = n - bb

                    if debug:
                        print(f"\t(7x{a} x 5x{b}) remain:{remain}")

                    # 割り切れた
                    if remain == 0:
                        return [a, b, 0, 0]

                    # 余った数で続きをやる

                    for c in reversed(range(0,101)):
                        cc = bb * 3 ** c

                        if cc <= n:
                            remain = n - cc

                            if debug:
                                print(f"\t\t(7x{a} x 5x{b} x 3x{c}) remain:{remain}")

                            # 割り切れた
                            if remain == 0:
                                return [a, b, c, 0]

                            # 余った数で続きをやる

                            for d in reversed(range(0,101)):
                                dd = cc * 2 ** d

                                if dd <= n:
                                    remain = n - dd

                                    if debug:
                                        print(f"\t\t\t(7x{a} x 5x{b} x 3x{c} x 2x{d}) remain:{remain}")

                                    # 割り切れた
                                    if remain == 0:
                                        return [a, b, c, d]


if __name__ == "__main__":

    print("""\
ある正の整数 n があるとする。
とりあえず n は 1～100 ぐらいを考えておく。

n を　2, 3, 5, 7　の素数で素因数分解する。

答えがいくつかあるとき、大きな数を多く使ったものを　１つ　選ぶことにする。
""")

    print("Please input number:")
    n = int(input())

    print(f"Anser:{factorize(n)}")

「　👆　コードが一気に短くなったな」

次の問い

N=1 Answer=[0, 0, 0, 0]
N=2 Answer=[0, 0, 0, 1]
N=3 Answer=[0, 0, 1, 0]
N=4 Answer=[0, 0, 0, 2]
N=5 Answer=[0, 1, 0, 0]
N=6 Answer=[0, 0, 1, 1]
N=7 Answer=[1, 0, 0, 0]
N=8 Answer=[0, 0, 0, 3]
N=9 Answer=[0, 0, 2, 0]
N=10 Answer=[0, 1, 0, 1]
N=11 Answer=None
N=12 Answer=[0, 0, 1, 2]
N=13 Answer=None
N=14 Answer=[1, 0, 0, 1]
N=15 Answer=[0, 1, 1, 0]
N=16 Answer=[0, 0, 0, 4]
N=17 Answer=None
N=18 Answer=[0, 0, 2, 1]
N=19 Answer=None
N=20 Answer=[0, 1, 0, 2]
N=21 Answer=[1, 0, 1, 0]
N=22 Answer=None
N=23 Answer=None
N=24 Answer=[0, 0, 1, 3]
N=25 Answer=[0, 2, 0, 0]
N=26 Answer=None
N=27 Answer=[0, 0, 3, 0]
N=28 Answer=[1, 0, 0, 2]
N=29 Answer=None
N=30 Answer=[0, 1, 1, 1]
N=31 Answer=None
N=32 Answer=[0, 0, 0, 5]
N=33 Answer=None
N=34 Answer=None
N=35 Answer=[1, 1, 0, 0]
N=36 Answer=[0, 0, 2, 2]
N=37 Answer=None
N=38 Answer=None
N=39 Answer=None
N=40 Answer=[0, 1, 0, 3]
N=41 Answer=None
N=42 Answer=[1, 0, 1, 1]
N=43 Answer=None
N=44 Answer=None
N=45 Answer=[0, 1, 2, 0]
N=46 Answer=None
N=47 Answer=None
N=48 Answer=[0, 0, 1, 4]
N=49 Answer=[2, 0, 0, 0]
N=50 Answer=[0, 2, 0, 1]
N=51 Answer=None
N=52 Answer=None
N=53 Answer=None
N=54 Answer=[0, 0, 3, 1]
N=55 Answer=None
N=56 Answer=[1, 0, 0, 3]
N=57 Answer=None
N=58 Answer=None
N=59 Answer=None
N=60 Answer=[0, 1, 1, 2]
N=61 Answer=None
N=62 Answer=None
N=63 Answer=[1, 0, 2, 0]
N=64 Answer=[0, 0, 0, 6]
N=65 Answer=None
N=66 Answer=None
N=67 Answer=None
N=68 Answer=None
N=69 Answer=None
N=70 Answer=[1, 1, 0, 1]
N=71 Answer=None
N=72 Answer=[0, 0, 2, 3]
N=73 Answer=None
N=74 Answer=None
N=75 Answer=[0, 2, 1, 0]
N=76 Answer=None
N=77 Answer=None
N=78 Answer=None
N=79 Answer=None
N=80 Answer=[0, 1, 0, 4]
N=81 Answer=[0, 0, 4, 0]
N=82 Answer=None
N=83 Answer=None
N=84 Answer=[1, 0, 1, 2]
N=85 Answer=None
N=86 Answer=None
N=87 Answer=None
N=88 Answer=None
N=89 Answer=None
N=90 Answer=[0, 1, 2, 1]
N=91 Answer=None
N=92 Answer=None
N=93 Answer=None
N=94 Answer=None
N=95 Answer=None
N=96 Answer=[0, 0, 1, 5]
N=97 Answer=None
N=98 Answer=[2, 0, 0, 1]
N=99 Answer=None
N=100 Answer=[0, 2, 0, 2]

「　👆　上図を見ると、 [0, 0, 0, 5]　は　[0, 1, 1, 1]　より　n が２個分強いんだが、
素因数分解を使わずに
[0, 0, 0, 5]　は　[0, 1, 1, 1]　より　n が２個分強いということを
計算する方法って　ある？」

「　5*3*2+2 = 2^5　だから、素因数から　n　を計算することはできるけど、」

「　そんな計算したくないわね。無いということにして　早く楽したいわね……」

「　もっと　よく見たいので　n = 101 ～ 1000 まで　見てみようぜ？」

N=101 Answer=None
N=102 Answer=None
N=103 Answer=None
N=104 Answer=None
N=105 Answer=[1, 1, 1, 0]
N=106 Answer=None
N=107 Answer=None
N=108 Answer=[0, 0, 3, 2]
N=109 Answer=None
N=110 Answer=None
N=111 Answer=None
N=112 Answer=[1, 0, 0, 4]
N=113 Answer=None
N=114 Answer=None
N=115 Answer=None
N=116 Answer=None
N=117 Answer=None
N=118 Answer=None
N=119 Answer=None
N=120 Answer=[0, 1, 1, 3]
N=121 Answer=None
N=122 Answer=None
N=123 Answer=None
N=124 Answer=None
N=125 Answer=[0, 3, 0, 0]
N=126 Answer=[1, 0, 2, 1]
N=127 Answer=None
N=128 Answer=[0, 0, 0, 7]
N=129 Answer=None
N=130 Answer=None
N=131 Answer=None
N=132 Answer=None
N=133 Answer=None
N=134 Answer=None
N=135 Answer=[0, 1, 3, 0]
N=136 Answer=None
N=137 Answer=None
N=138 Answer=None
N=139 Answer=None
N=140 Answer=[1, 1, 0, 2]
N=141 Answer=None
N=142 Answer=None
N=143 Answer=None
N=144 Answer=[0, 0, 2, 4]
N=145 Answer=None
N=146 Answer=None
N=147 Answer=[2, 0, 1, 0]
N=148 Answer=None
N=149 Answer=None
N=150 Answer=[0, 2, 1, 1]
N=151 Answer=None
N=152 Answer=None
N=153 Answer=None
N=154 Answer=None
N=155 Answer=None
N=156 Answer=None
N=157 Answer=None
N=158 Answer=None
N=159 Answer=None
N=160 Answer=[0, 1, 0, 5]
N=161 Answer=None
N=162 Answer=[0, 0, 4, 1]
N=163 Answer=None
N=164 Answer=None
N=165 Answer=None
N=166 Answer=None
N=167 Answer=None
N=168 Answer=[1, 0, 1, 3]
N=169 Answer=None
N=170 Answer=None
N=171 Answer=None
N=172 Answer=None
N=173 Answer=None
N=174 Answer=None
N=175 Answer=[1, 2, 0, 0]
N=176 Answer=None
N=177 Answer=None
N=178 Answer=None
N=179 Answer=None
N=180 Answer=[0, 1, 2, 2]
N=181 Answer=None
N=182 Answer=None
N=183 Answer=None
N=184 Answer=None
N=185 Answer=None
N=186 Answer=None
N=187 Answer=None
N=188 Answer=None
N=189 Answer=[1, 0, 3, 0]
N=190 Answer=None
N=191 Answer=None
N=192 Answer=[0, 0, 1, 6]
N=193 Answer=None
N=194 Answer=None
N=195 Answer=None
N=196 Answer=[2, 0, 0, 2]
N=197 Answer=None
N=198 Answer=None
N=199 Answer=None
N=200 Answer=[0, 2, 0, 3]
N=201 Answer=None
N=202 Answer=None
N=203 Answer=None
N=204 Answer=None
N=205 Answer=None
N=206 Answer=None
N=207 Answer=None
N=208 Answer=None
N=209 Answer=None
N=210 Answer=[1, 1, 1, 1]
N=211 Answer=None
N=212 Answer=None
N=213 Answer=None
N=214 Answer=None
N=215 Answer=None
N=216 Answer=[0, 0, 3, 3]
N=217 Answer=None
N=218 Answer=None
N=219 Answer=None
N=220 Answer=None
N=221 Answer=None
N=222 Answer=None
N=223 Answer=None
N=224 Answer=[1, 0, 0, 5]
N=225 Answer=[0, 2, 2, 0]
N=226 Answer=None
N=227 Answer=None
N=228 Answer=None
N=229 Answer=None
N=230 Answer=None
N=231 Answer=None
N=232 Answer=None
N=233 Answer=None
N=234 Answer=None
N=235 Answer=None
N=236 Answer=None
N=237 Answer=None
N=238 Answer=None
N=239 Answer=None
N=240 Answer=[0, 1, 1, 4]
N=241 Answer=None
N=242 Answer=None
N=243 Answer=[0, 0, 5, 0]
N=244 Answer=None
N=245 Answer=[2, 1, 0, 0]
N=246 Answer=None
N=247 Answer=None
N=248 Answer=None
N=249 Answer=None
N=250 Answer=[0, 3, 0, 1]
N=251 Answer=None
N=252 Answer=[1, 0, 2, 2]
N=253 Answer=None
N=254 Answer=None
N=255 Answer=None
N=256 Answer=[0, 0, 0, 8]
N=257 Answer=None
N=258 Answer=None
N=259 Answer=None
N=260 Answer=None
N=261 Answer=None
N=262 Answer=None
N=263 Answer=None
N=264 Answer=None
N=265 Answer=None
N=266 Answer=None
N=267 Answer=None
N=268 Answer=None
N=269 Answer=None
N=270 Answer=[0, 1, 3, 1]
N=271 Answer=None
N=272 Answer=None
N=273 Answer=None
N=274 Answer=None
N=275 Answer=None
N=276 Answer=None
N=277 Answer=None
N=278 Answer=None
N=279 Answer=None
N=280 Answer=[1, 1, 0, 3]
N=281 Answer=None
N=282 Answer=None
N=283 Answer=None
N=284 Answer=None
N=285 Answer=None
N=286 Answer=None
N=287 Answer=None
N=288 Answer=[0, 0, 2, 5]
N=289 Answer=None
N=290 Answer=None
N=291 Answer=None
N=292 Answer=None
N=293 Answer=None
N=294 Answer=[2, 0, 1, 1]
N=295 Answer=None
N=296 Answer=None
N=297 Answer=None
N=298 Answer=None
N=299 Answer=None
N=300 Answer=[0, 2, 1, 2]
N=301 Answer=None
N=302 Answer=None
N=303 Answer=None
N=304 Answer=None
N=305 Answer=None
N=306 Answer=None
N=307 Answer=None
N=308 Answer=None
N=309 Answer=None
N=310 Answer=None
N=311 Answer=None
N=312 Answer=None
N=313 Answer=None
N=314 Answer=None
N=315 Answer=[1, 1, 2, 0]
N=316 Answer=None
N=317 Answer=None
N=318 Answer=None
N=319 Answer=None
N=320 Answer=[0, 1, 0, 6]
N=321 Answer=None
N=322 Answer=None
N=323 Answer=None
N=324 Answer=[0, 0, 4, 2]
N=325 Answer=None
N=326 Answer=None
N=327 Answer=None
N=328 Answer=None
N=329 Answer=None
N=330 Answer=None
N=331 Answer=None
N=332 Answer=None
N=333 Answer=None
N=334 Answer=None
N=335 Answer=None
N=336 Answer=[1, 0, 1, 4]
N=337 Answer=None
N=338 Answer=None
N=339 Answer=None
N=340 Answer=None
N=341 Answer=None
N=342 Answer=None
N=343 Answer=[3, 0, 0, 0]
N=344 Answer=None
N=345 Answer=None
N=346 Answer=None
N=347 Answer=None
N=348 Answer=None
N=349 Answer=None
N=350 Answer=[1, 2, 0, 1]
N=351 Answer=None
N=352 Answer=None
N=353 Answer=None
N=354 Answer=None
N=355 Answer=None
N=356 Answer=None
N=357 Answer=None
N=358 Answer=None
N=359 Answer=None
N=360 Answer=[0, 1, 2, 3]
N=361 Answer=None
N=362 Answer=None
N=363 Answer=None
N=364 Answer=None
N=365 Answer=None
N=366 Answer=None
N=367 Answer=None
N=368 Answer=None
N=369 Answer=None
N=370 Answer=None
N=371 Answer=None
N=372 Answer=None
N=373 Answer=None
N=374 Answer=None
N=375 Answer=[0, 3, 1, 0]
N=376 Answer=None
N=377 Answer=None
N=378 Answer=[1, 0, 3, 1]
N=379 Answer=None
N=380 Answer=None
N=381 Answer=None
N=382 Answer=None
N=383 Answer=None
N=384 Answer=[0, 0, 1, 7]
N=385 Answer=None
N=386 Answer=None
N=387 Answer=None
N=388 Answer=None
N=389 Answer=None
N=390 Answer=None
N=391 Answer=None
N=392 Answer=[2, 0, 0, 3]
N=393 Answer=None
N=394 Answer=None
N=395 Answer=None
N=396 Answer=None
N=397 Answer=None
N=398 Answer=None
N=399 Answer=None
N=400 Answer=[0, 2, 0, 4]
N=401 Answer=None
N=402 Answer=None
N=403 Answer=None
N=404 Answer=None
N=405 Answer=[0, 1, 4, 0]
N=406 Answer=None
N=407 Answer=None
N=408 Answer=None
N=409 Answer=None
N=410 Answer=None
N=411 Answer=None
N=412 Answer=None
N=413 Answer=None
N=414 Answer=None
N=415 Answer=None
N=416 Answer=None
N=417 Answer=None
N=418 Answer=None
N=419 Answer=None
N=420 Answer=[1, 1, 1, 2]
N=421 Answer=None
N=422 Answer=None
N=423 Answer=None
N=424 Answer=None
N=425 Answer=None
N=426 Answer=None
N=427 Answer=None
N=428 Answer=None
N=429 Answer=None
N=430 Answer=None
N=431 Answer=None
N=432 Answer=[0, 0, 3, 4]
N=433 Answer=None
N=434 Answer=None
N=435 Answer=None
N=436 Answer=None
N=437 Answer=None
N=438 Answer=None
N=439 Answer=None
N=440 Answer=None
N=441 Answer=[2, 0, 2, 0]
N=442 Answer=None
N=443 Answer=None
N=444 Answer=None
N=445 Answer=None
N=446 Answer=None
N=447 Answer=None
N=448 Answer=[1, 0, 0, 6]
N=449 Answer=None
N=450 Answer=[0, 2, 2, 1]
N=451 Answer=None
N=452 Answer=None
N=453 Answer=None
N=454 Answer=None
N=455 Answer=None
N=456 Answer=None
N=457 Answer=None
N=458 Answer=None
N=459 Answer=None
N=460 Answer=None
N=461 Answer=None
N=462 Answer=None
N=463 Answer=None
N=464 Answer=None
N=465 Answer=None
N=466 Answer=None
N=467 Answer=None
N=468 Answer=None
N=469 Answer=None
N=470 Answer=None
N=471 Answer=None
N=472 Answer=None
N=473 Answer=None
N=474 Answer=None
N=475 Answer=None
N=476 Answer=None
N=477 Answer=None
N=478 Answer=None
N=479 Answer=None
N=480 Answer=[0, 1, 1, 5]
N=481 Answer=None
N=482 Answer=None
N=483 Answer=None
N=484 Answer=None
N=485 Answer=None
N=486 Answer=[0, 0, 5, 1]
N=487 Answer=None
N=488 Answer=None
N=489 Answer=None
N=490 Answer=[2, 1, 0, 1]
N=491 Answer=None
N=492 Answer=None
N=493 Answer=None
N=494 Answer=None
N=495 Answer=None
N=496 Answer=None
N=497 Answer=None
N=498 Answer=None
N=499 Answer=None
N=500 Answer=[0, 3, 0, 2]
N=501 Answer=None
N=502 Answer=None
N=503 Answer=None
N=504 Answer=[1, 0, 2, 3]
N=505 Answer=None
N=506 Answer=None
N=507 Answer=None
N=508 Answer=None
N=509 Answer=None
N=510 Answer=None
N=511 Answer=None
N=512 Answer=[0, 0, 0, 9]
N=513 Answer=None
N=514 Answer=None
N=515 Answer=None
N=516 Answer=None
N=517 Answer=None
N=518 Answer=None
N=519 Answer=None
N=520 Answer=None
N=521 Answer=None
N=522 Answer=None
N=523 Answer=None
N=524 Answer=None
N=525 Answer=[1, 2, 1, 0]
N=526 Answer=None
N=527 Answer=None
N=528 Answer=None
N=529 Answer=None
N=530 Answer=None
N=531 Answer=None
N=532 Answer=None
N=533 Answer=None
N=534 Answer=None
N=535 Answer=None
N=536 Answer=None
N=537 Answer=None
N=538 Answer=None
N=539 Answer=None
N=540 Answer=[0, 1, 3, 2]
N=541 Answer=None
N=542 Answer=None
N=543 Answer=None
N=544 Answer=None
N=545 Answer=None
N=546 Answer=None
N=547 Answer=None
N=548 Answer=None
N=549 Answer=None
N=550 Answer=None
N=551 Answer=None
N=552 Answer=None
N=553 Answer=None
N=554 Answer=None
N=555 Answer=None
N=556 Answer=None
N=557 Answer=None
N=558 Answer=None
N=559 Answer=None
N=560 Answer=[1, 1, 0, 4]
N=561 Answer=None
N=562 Answer=None
N=563 Answer=None
N=564 Answer=None
N=565 Answer=None
N=566 Answer=None
N=567 Answer=[1, 0, 4, 0]
N=568 Answer=None
N=569 Answer=None
N=570 Answer=None
N=571 Answer=None
N=572 Answer=None
N=573 Answer=None
N=574 Answer=None
N=575 Answer=None
N=576 Answer=[0, 0, 2, 6]
N=577 Answer=None
N=578 Answer=None
N=579 Answer=None
N=580 Answer=None
N=581 Answer=None
N=582 Answer=None
N=583 Answer=None
N=584 Answer=None
N=585 Answer=None
N=586 Answer=None
N=587 Answer=None
N=588 Answer=[2, 0, 1, 2]
N=589 Answer=None
N=590 Answer=None
N=591 Answer=None
N=592 Answer=None
N=593 Answer=None
N=594 Answer=None
N=595 Answer=None
N=596 Answer=None
N=597 Answer=None
N=598 Answer=None
N=599 Answer=None
N=600 Answer=[0, 2, 1, 3]
N=601 Answer=None
N=602 Answer=None
N=603 Answer=None
N=604 Answer=None
N=605 Answer=None
N=606 Answer=None
N=607 Answer=None
N=608 Answer=None
N=609 Answer=None
N=610 Answer=None
N=611 Answer=None
N=612 Answer=None
N=613 Answer=None
N=614 Answer=None
N=615 Answer=None
N=616 Answer=None
N=617 Answer=None
N=618 Answer=None
N=619 Answer=None
N=620 Answer=None
N=621 Answer=None
N=622 Answer=None
N=623 Answer=None
N=624 Answer=None
N=625 Answer=[0, 4, 0, 0]
N=626 Answer=None
N=627 Answer=None
N=628 Answer=None
N=629 Answer=None
N=630 Answer=[1, 1, 2, 1]
N=631 Answer=None
N=632 Answer=None
N=633 Answer=None
N=634 Answer=None
N=635 Answer=None
N=636 Answer=None
N=637 Answer=None
N=638 Answer=None
N=639 Answer=None
N=640 Answer=[0, 1, 0, 7]
N=641 Answer=None
N=642 Answer=None
N=643 Answer=None
N=644 Answer=None
N=645 Answer=None
N=646 Answer=None
N=647 Answer=None
N=648 Answer=[0, 0, 4, 3]
N=649 Answer=None
N=650 Answer=None
N=651 Answer=None
N=652 Answer=None
N=653 Answer=None
N=654 Answer=None
N=655 Answer=None
N=656 Answer=None
N=657 Answer=None
N=658 Answer=None
N=659 Answer=None
N=660 Answer=None
N=661 Answer=None
N=662 Answer=None
N=663 Answer=None
N=664 Answer=None
N=665 Answer=None
N=666 Answer=None
N=667 Answer=None
N=668 Answer=None
N=669 Answer=None
N=670 Answer=None
N=671 Answer=None
N=672 Answer=[1, 0, 1, 5]
N=673 Answer=None
N=674 Answer=None
N=675 Answer=[0, 2, 3, 0]
N=676 Answer=None
N=677 Answer=None
N=678 Answer=None
N=679 Answer=None
N=680 Answer=None
N=681 Answer=None
N=682 Answer=None
N=683 Answer=None
N=684 Answer=None
N=685 Answer=None
N=686 Answer=[3, 0, 0, 1]
N=687 Answer=None
N=688 Answer=None
N=689 Answer=None
N=690 Answer=None
N=691 Answer=None
N=692 Answer=None
N=693 Answer=None
N=694 Answer=None
N=695 Answer=None
N=696 Answer=None
N=697 Answer=None
N=698 Answer=None
N=699 Answer=None
N=700 Answer=[1, 2, 0, 2]
N=701 Answer=None
N=702 Answer=None
N=703 Answer=None
N=704 Answer=None
N=705 Answer=None
N=706 Answer=None
N=707 Answer=None
N=708 Answer=None
N=709 Answer=None
N=710 Answer=None
N=711 Answer=None
N=712 Answer=None
N=713 Answer=None
N=714 Answer=None
N=715 Answer=None
N=716 Answer=None
N=717 Answer=None
N=718 Answer=None
N=719 Answer=None
N=720 Answer=[0, 1, 2, 4]
N=721 Answer=None
N=722 Answer=None
N=723 Answer=None
N=724 Answer=None
N=725 Answer=None
N=726 Answer=None
N=727 Answer=None
N=728 Answer=None
N=729 Answer=[0, 0, 6, 0]
N=730 Answer=None
N=731 Answer=None
N=732 Answer=None
N=733 Answer=None
N=734 Answer=None
N=735 Answer=[2, 1, 1, 0]
N=736 Answer=None
N=737 Answer=None
N=738 Answer=None
N=739 Answer=None
N=740 Answer=None
N=741 Answer=None
N=742 Answer=None
N=743 Answer=None
N=744 Answer=None
N=745 Answer=None
N=746 Answer=None
N=747 Answer=None
N=748 Answer=None
N=749 Answer=None
N=750 Answer=[0, 3, 1, 1]
N=751 Answer=None
N=752 Answer=None
N=753 Answer=None
N=754 Answer=None
N=755 Answer=None
N=756 Answer=[1, 0, 3, 2]
N=757 Answer=None
N=758 Answer=None
N=759 Answer=None
N=760 Answer=None
N=761 Answer=None
N=762 Answer=None
N=763 Answer=None
N=764 Answer=None
N=765 Answer=None
N=766 Answer=None
N=767 Answer=None
N=768 Answer=[0, 0, 1, 8]
N=769 Answer=None
N=770 Answer=None
N=771 Answer=None
N=772 Answer=None
N=773 Answer=None
N=774 Answer=None
N=775 Answer=None
N=776 Answer=None
N=777 Answer=None
N=778 Answer=None
N=779 Answer=None
N=780 Answer=None
N=781 Answer=None
N=782 Answer=None
N=783 Answer=None
N=784 Answer=[2, 0, 0, 4]
N=785 Answer=None
N=786 Answer=None
N=787 Answer=None
N=788 Answer=None
N=789 Answer=None
N=790 Answer=None
N=791 Answer=None
N=792 Answer=None
N=793 Answer=None
N=794 Answer=None
N=795 Answer=None
N=796 Answer=None
N=797 Answer=None
N=798 Answer=None
N=799 Answer=None
N=800 Answer=[0, 2, 0, 5]
N=801 Answer=None
N=802 Answer=None
N=803 Answer=None
N=804 Answer=None
N=805 Answer=None
N=806 Answer=None
N=807 Answer=None
N=808 Answer=None
N=809 Answer=None
N=810 Answer=[0, 1, 4, 1]
N=811 Answer=None
N=812 Answer=None
N=813 Answer=None
N=814 Answer=None
N=815 Answer=None
N=816 Answer=None
N=817 Answer=None
N=818 Answer=None
N=819 Answer=None
N=820 Answer=None
N=821 Answer=None
N=822 Answer=None
N=823 Answer=None
N=824 Answer=None
N=825 Answer=None
N=826 Answer=None
N=827 Answer=None
N=828 Answer=None
N=829 Answer=None
N=830 Answer=None
N=831 Answer=None
N=832 Answer=None
N=833 Answer=None
N=834 Answer=None
N=835 Answer=None
N=836 Answer=None
N=837 Answer=None
N=838 Answer=None
N=839 Answer=None
N=840 Answer=[1, 1, 1, 3]
N=841 Answer=None
N=842 Answer=None
N=843 Answer=None
N=844 Answer=None
N=845 Answer=None
N=846 Answer=None
N=847 Answer=None
N=848 Answer=None
N=849 Answer=None
N=850 Answer=None
N=851 Answer=None
N=852 Answer=None
N=853 Answer=None
N=854 Answer=None
N=855 Answer=None
N=856 Answer=None
N=857 Answer=None
N=858 Answer=None
N=859 Answer=None
N=860 Answer=None
N=861 Answer=None
N=862 Answer=None
N=863 Answer=None
N=864 Answer=[0, 0, 3, 5]
N=865 Answer=None
N=866 Answer=None
N=867 Answer=None
N=868 Answer=None
N=869 Answer=None
N=870 Answer=None
N=871 Answer=None
N=872 Answer=None
N=873 Answer=None
N=874 Answer=None
N=875 Answer=[1, 3, 0, 0]
N=876 Answer=None
N=877 Answer=None
N=878 Answer=None
N=879 Answer=None
N=880 Answer=None
N=881 Answer=None
N=882 Answer=[2, 0, 2, 1]
N=883 Answer=None
N=884 Answer=None
N=885 Answer=None
N=886 Answer=None
N=887 Answer=None
N=888 Answer=None
N=889 Answer=None
N=890 Answer=None
N=891 Answer=None
N=892 Answer=None
N=893 Answer=None
N=894 Answer=None
N=895 Answer=None
N=896 Answer=[1, 0, 0, 7]
N=897 Answer=None
N=898 Answer=None
N=899 Answer=None
N=900 Answer=[0, 2, 2, 2]
N=901 Answer=None
N=902 Answer=None
N=903 Answer=None
N=904 Answer=None
N=905 Answer=None
N=906 Answer=None
N=907 Answer=None
N=908 Answer=None
N=909 Answer=None
N=910 Answer=None
N=911 Answer=None
N=912 Answer=None
N=913 Answer=None
N=914 Answer=None
N=915 Answer=None
N=916 Answer=None
N=917 Answer=None
N=918 Answer=None
N=919 Answer=None
N=920 Answer=None
N=921 Answer=None
N=922 Answer=None
N=923 Answer=None
N=924 Answer=None
N=925 Answer=None
N=926 Answer=None
N=927 Answer=None
N=928 Answer=None
N=929 Answer=None
N=930 Answer=None
N=931 Answer=None
N=932 Answer=None
N=933 Answer=None
N=934 Answer=None
N=935 Answer=None
N=936 Answer=None
N=937 Answer=None
N=938 Answer=None
N=939 Answer=None
N=940 Answer=None
N=941 Answer=None
N=942 Answer=None
N=943 Answer=None
N=944 Answer=None
N=945 Answer=[1, 1, 3, 0]
N=946 Answer=None
N=947 Answer=None
N=948 Answer=None
N=949 Answer=None
N=950 Answer=None
N=951 Answer=None
N=952 Answer=None
N=953 Answer=None
N=954 Answer=None
N=955 Answer=None
N=956 Answer=None
N=957 Answer=None
N=958 Answer=None
N=959 Answer=None
N=960 Answer=[0, 1, 1, 6]
N=961 Answer=None
N=962 Answer=None
N=963 Answer=None
N=964 Answer=None
N=965 Answer=None
N=966 Answer=None
N=967 Answer=None
N=968 Answer=None
N=969 Answer=None
N=970 Answer=None
N=971 Answer=None
N=972 Answer=[0, 0, 5, 2]
N=973 Answer=None
N=974 Answer=None
N=975 Answer=None
N=976 Answer=None
N=977 Answer=None
N=978 Answer=None
N=979 Answer=None
N=980 Answer=[2, 1, 0, 2]
N=981 Answer=None
N=982 Answer=None
N=983 Answer=None
N=984 Answer=None
N=985 Answer=None
N=986 Answer=None
N=987 Answer=None
N=988 Answer=None
N=989 Answer=None
N=990 Answer=None
N=991 Answer=None
N=992 Answer=None
N=993 Answer=None
N=994 Answer=None
N=995 Answer=None
N=996 Answer=None
N=997 Answer=None
N=998 Answer=None
N=999 Answer=None
N=1000 Answer=[0, 3, 0, 3]

「　👆　満足かだぜ、お父ん？」

「　過疎化してくんだなあ」

割り切れる数だけ並べるとどうなる？

「　割り切れる数だけ並べるとどうなるんだぜ？」

(0) n=1 Answer=[0, 0, 0, 0]
(1) n=2 Answer=[0, 0, 0, 1]
(2) n=3 Answer=[0, 0, 1, 0]
(3) n=4 Answer=[0, 0, 0, 2]
(4) n=5 Answer=[0, 1, 0, 0]
(5) n=6 Answer=[0, 0, 1, 1]
(6) n=7 Answer=[1, 0, 0, 0]
(7) n=8 Answer=[0, 0, 0, 3]
(8) n=9 Answer=[0, 0, 2, 0]
(9) n=10 Answer=[0, 1, 0, 1]
(10) n=12 Answer=[0, 0, 1, 2]
(11) n=14 Answer=[1, 0, 0, 1]
(12) n=15 Answer=[0, 1, 1, 0]
(13) n=16 Answer=[0, 0, 0, 4]
(14) n=18 Answer=[0, 0, 2, 1]
(15) n=20 Answer=[0, 1, 0, 2]
(16) n=21 Answer=[1, 0, 1, 0]
(17) n=24 Answer=[0, 0, 1, 3]
(18) n=25 Answer=[0, 2, 0, 0]
(19) n=27 Answer=[0, 0, 3, 0]
(20) n=28 Answer=[1, 0, 0, 2]
(21) n=30 Answer=[0, 1, 1, 1]
(22) n=32 Answer=[0, 0, 0, 5]
(23) n=35 Answer=[1, 1, 0, 0]
(24) n=36 Answer=[0, 0, 2, 2]
(25) n=40 Answer=[0, 1, 0, 3]
(26) n=42 Answer=[1, 0, 1, 1]
(27) n=45 Answer=[0, 1, 2, 0]
(28) n=48 Answer=[0, 0, 1, 4]
(29) n=49 Answer=[2, 0, 0, 0]
(30) n=50 Answer=[0, 2, 0, 1]
(31) n=54 Answer=[0, 0, 3, 1]
(32) n=56 Answer=[1, 0, 0, 3]
(33) n=60 Answer=[0, 1, 1, 2]
(34) n=63 Answer=[1, 0, 2, 0]
(35) n=64 Answer=[0, 0, 0, 6]
(36) n=70 Answer=[1, 1, 0, 1]
(37) n=72 Answer=[0, 0, 2, 3]
(38) n=75 Answer=[0, 2, 1, 0]
(39) n=80 Answer=[0, 1, 0, 4]
(40) n=81 Answer=[0, 0, 4, 0]
(41) n=84 Answer=[1, 0, 1, 2]
(42) n=90 Answer=[0, 1, 2, 1]
(43) n=96 Answer=[0, 0, 1, 5]
(44) n=98 Answer=[2, 0, 0, 1]
(45) n=100 Answer=[0, 2, 0, 2]
(46) n=105 Answer=[1, 1, 1, 0]
(47) n=108 Answer=[0, 0, 3, 2]
(48) n=112 Answer=[1, 0, 0, 4]
(49) n=120 Answer=[0, 1, 1, 3]
(50) n=125 Answer=[0, 3, 0, 0]
(51) n=126 Answer=[1, 0, 2, 1]
(52) n=128 Answer=[0, 0, 0, 7]
(53) n=135 Answer=[0, 1, 3, 0]
(54) n=140 Answer=[1, 1, 0, 2]
(55) n=144 Answer=[0, 0, 2, 4]
(56) n=147 Answer=[2, 0, 1, 0]
(57) n=150 Answer=[0, 2, 1, 1]
(58) n=160 Answer=[0, 1, 0, 5]
(59) n=162 Answer=[0, 0, 4, 1]
(60) n=168 Answer=[1, 0, 1, 3]
(61) n=175 Answer=[1, 2, 0, 0]
(62) n=180 Answer=[0, 1, 2, 2]
(63) n=189 Answer=[1, 0, 3, 0]
(64) n=192 Answer=[0, 0, 1, 6]
(65) n=196 Answer=[2, 0, 0, 2]
(66) n=200 Answer=[0, 2, 0, 3]
(67) n=210 Answer=[1, 1, 1, 1]
(68) n=216 Answer=[0, 0, 3, 3]
(69) n=224 Answer=[1, 0, 0, 5]
(70) n=225 Answer=[0, 2, 2, 0]
(71) n=240 Answer=[0, 1, 1, 4]
(72) n=243 Answer=[0, 0, 5, 0]
(73) n=245 Answer=[2, 1, 0, 0]
(74) n=250 Answer=[0, 3, 0, 1]
(75) n=252 Answer=[1, 0, 2, 2]
(76) n=256 Answer=[0, 0, 0, 8]
(77) n=270 Answer=[0, 1, 3, 1]
(78) n=280 Answer=[1, 1, 0, 3]
(79) n=288 Answer=[0, 0, 2, 5]
(80) n=294 Answer=[2, 0, 1, 1]
(81) n=300 Answer=[0, 2, 1, 2]
(82) n=315 Answer=[1, 1, 2, 0]
(83) n=320 Answer=[0, 1, 0, 6]
(84) n=324 Answer=[0, 0, 4, 2]
(85) n=336 Answer=[1, 0, 1, 4]
(86) n=343 Answer=[3, 0, 0, 0]
(87) n=350 Answer=[1, 2, 0, 1]
(88) n=360 Answer=[0, 1, 2, 3]
(89) n=375 Answer=[0, 3, 1, 0]
(90) n=378 Answer=[1, 0, 3, 1]
(91) n=384 Answer=[0, 0, 1, 7]
(92) n=392 Answer=[2, 0, 0, 3]
(93) n=400 Answer=[0, 2, 0, 4]
(94) n=405 Answer=[0, 1, 4, 0]
(95) n=420 Answer=[1, 1, 1, 2]
(96) n=432 Answer=[0, 0, 3, 4]
(97) n=441 Answer=[2, 0, 2, 0]
(98) n=448 Answer=[1, 0, 0, 6]
(99) n=450 Answer=[0, 2, 2, 1]
(100) n=480 Answer=[0, 1, 1, 5]
(101) n=486 Answer=[0, 0, 5, 1]
(102) n=490 Answer=[2, 1, 0, 1]
(103) n=500 Answer=[0, 3, 0, 2]
(104) n=504 Answer=[1, 0, 2, 3]
(105) n=512 Answer=[0, 0, 0, 9]
(106) n=525 Answer=[1, 2, 1, 0]
(107) n=540 Answer=[0, 1, 3, 2]
(108) n=560 Answer=[1, 1, 0, 4]
(109) n=567 Answer=[1, 0, 4, 0]
(110) n=576 Answer=[0, 0, 2, 6]
(111) n=588 Answer=[2, 0, 1, 2]
(112) n=600 Answer=[0, 2, 1, 3]
(113) n=625 Answer=[0, 4, 0, 0]
(114) n=630 Answer=[1, 1, 2, 1]
(115) n=640 Answer=[0, 1, 0, 7]
(116) n=648 Answer=[0, 0, 4, 3]
(117) n=672 Answer=[1, 0, 1, 5]
(118) n=675 Answer=[0, 2, 3, 0]
(119) n=686 Answer=[3, 0, 0, 1]
(120) n=700 Answer=[1, 2, 0, 2]
(121) n=720 Answer=[0, 1, 2, 4]
(122) n=729 Answer=[0, 0, 6, 0]
(123) n=735 Answer=[2, 1, 1, 0]
(124) n=750 Answer=[0, 3, 1, 1]
(125) n=756 Answer=[1, 0, 3, 2]
(126) n=768 Answer=[0, 0, 1, 8]
(127) n=784 Answer=[2, 0, 0, 4]
(128) n=800 Answer=[0, 2, 0, 5]
(129) n=810 Answer=[0, 1, 4, 1]
(130) n=840 Answer=[1, 1, 1, 3]
(131) n=864 Answer=[0, 0, 3, 5]
(132) n=875 Answer=[1, 3, 0, 0]
(133) n=882 Answer=[2, 0, 2, 1]
(134) n=896 Answer=[1, 0, 0, 7]
(135) n=900 Answer=[0, 2, 2, 2]
(136) n=945 Answer=[1, 1, 3, 0]
(137) n=960 Answer=[0, 1, 1, 6]
(138) n=972 Answer=[0, 0, 5, 2]
(139) n=980 Answer=[2, 1, 0, 2]
(140) n=1000 Answer=[0, 3, 0, 3]
(141) n=1008 Answer=[1, 0, 2, 4]
(142) n=1024 Answer=[0, 0, 0, 10]
(143) n=1029 Answer=[3, 0, 1, 0]
(144) n=1050 Answer=[1, 2, 1, 1]
(145) n=1080 Answer=[0, 1, 3, 3]
(146) n=1120 Answer=[1, 1, 0, 5]
(147) n=1125 Answer=[0, 3, 2, 0]
(148) n=1134 Answer=[1, 0, 4, 1]
(149) n=1152 Answer=[0, 0, 2, 7]
(150) n=1176 Answer=[2, 0, 1, 3]
(151) n=1200 Answer=[0, 2, 1, 4]
(152) n=1215 Answer=[0, 1, 5, 0]
(153) n=1225 Answer=[2, 2, 0, 0]
(154) n=1250 Answer=[0, 4, 0, 1]
(155) n=1260 Answer=[1, 1, 2, 2]
(156) n=1280 Answer=[0, 1, 0, 8]
(157) n=1296 Answer=[0, 0, 4, 4]
(158) n=1323 Answer=[2, 0, 3, 0]
(159) n=1344 Answer=[1, 0, 1, 6]
(160) n=1350 Answer=[0, 2, 3, 1]
(161) n=1372 Answer=[3, 0, 0, 2]
(162) n=1400 Answer=[1, 2, 0, 3]
(163) n=1440 Answer=[0, 1, 2, 5]
(164) n=1458 Answer=[0, 0, 6, 1]
(165) n=1470 Answer=[2, 1, 1, 1]
(166) n=1500 Answer=[0, 3, 1, 2]
(167) n=1512 Answer=[1, 0, 3, 3]
(168) n=1536 Answer=[0, 0, 1, 9]
(169) n=1568 Answer=[2, 0, 0, 5]
(170) n=1575 Answer=[1, 2, 2, 0]
(171) n=1600 Answer=[0, 2, 0, 6]
(172) n=1620 Answer=[0, 1, 4, 2]
(173) n=1680 Answer=[1, 1, 1, 4]
(174) n=1701 Answer=[1, 0, 5, 0]
(175) n=1715 Answer=[3, 1, 0, 0]
(176) n=1728 Answer=[0, 0, 3, 6]
(177) n=1750 Answer=[1, 3, 0, 1]
(178) n=1764 Answer=[2, 0, 2, 2]
(179) n=1792 Answer=[1, 0, 0, 8]
(180) n=1800 Answer=[0, 2, 2, 3]
(181) n=1875 Answer=[0, 4, 1, 0]
(182) n=1890 Answer=[1, 1, 3, 1]
(183) n=1920 Answer=[0, 1, 1, 7]
(184) n=1944 Answer=[0, 0, 5, 3]
(185) n=1960 Answer=[2, 1, 0, 3]
(186) n=2000 Answer=[0, 3, 0, 4]
(187) n=2016 Answer=[1, 0, 2, 5]
(188) n=2025 Answer=[0, 2, 4, 0]
(189) n=2048 Answer=[0, 0, 0, 11]
(190) n=2058 Answer=[3, 0, 1, 1]
(191) n=2100 Answer=[1, 2, 1, 2]
(192) n=2160 Answer=[0, 1, 3, 4]
(193) n=2187 Answer=[0, 0, 7, 0]
(194) n=2205 Answer=[2, 1, 2, 0]
(195) n=2240 Answer=[1, 1, 0, 6]
(196) n=2250 Answer=[0, 3, 2, 1]
(197) n=2268 Answer=[1, 0, 4, 2]
(198) n=2304 Answer=[0, 0, 2, 8]
(199) n=2352 Answer=[2, 0, 1, 4]
(200) n=2400 Answer=[0, 2, 1, 5]
(201) n=2401 Answer=[4, 0, 0, 0]
(202) n=2430 Answer=[0, 1, 5, 1]
(203) n=2450 Answer=[2, 2, 0, 1]
(204) n=2500 Answer=[0, 4, 0, 2]
(205) n=2520 Answer=[1, 1, 2, 3]
(206) n=2560 Answer=[0, 1, 0, 9]
(207) n=2592 Answer=[0, 0, 4, 5]
(208) n=2625 Answer=[1, 3, 1, 0]
(209) n=2646 Answer=[2, 0, 3, 1]
(210) n=2688 Answer=[1, 0, 1, 7]
(211) n=2700 Answer=[0, 2, 3, 2]
(212) n=2744 Answer=[3, 0, 0, 3]
(213) n=2800 Answer=[1, 2, 0, 4]
(214) n=2835 Answer=[1, 1, 4, 0]
(215) n=2880 Answer=[0, 1, 2, 6]
(216) n=2916 Answer=[0, 0, 6, 2]
(217) n=2940 Answer=[2, 1, 1, 2]
(218) n=3000 Answer=[0, 3, 1, 3]
(219) n=3024 Answer=[1, 0, 3, 4]
(220) n=3072 Answer=[0, 0, 1, 10]
(221) n=3087 Answer=[3, 0, 2, 0]
(222) n=3125 Answer=[0, 5, 0, 0]
(223) n=3136 Answer=[2, 0, 0, 6]
(224) n=3150 Answer=[1, 2, 2, 1]
(225) n=3200 Answer=[0, 2, 0, 7]
(226) n=3240 Answer=[0, 1, 4, 3]
(227) n=3360 Answer=[1, 1, 1, 5]
(228) n=3375 Answer=[0, 3, 3, 0]
(229) n=3402 Answer=[1, 0, 5, 1]
(230) n=3430 Answer=[3, 1, 0, 1]
(231) n=3456 Answer=[0, 0, 3, 7]
(232) n=3500 Answer=[1, 3, 0, 2]
(233) n=3528 Answer=[2, 0, 2, 3]
(234) n=3584 Answer=[1, 0, 0, 9]
(235) n=3600 Answer=[0, 2, 2, 4]
(236) n=3645 Answer=[0, 1, 6, 0]
(237) n=3675 Answer=[2, 2, 1, 0]
(238) n=3750 Answer=[0, 4, 1, 1]
(239) n=3780 Answer=[1, 1, 3, 2]
(240) n=3840 Answer=[0, 1, 1, 8]
(241) n=3888 Answer=[0, 0, 5, 4]
(242) n=3920 Answer=[2, 1, 0, 4]
(243) n=3969 Answer=[2, 0, 4, 0]
(244) n=4000 Answer=[0, 3, 0, 5]
(245) n=4032 Answer=[1, 0, 2, 6]
(246) n=4050 Answer=[0, 2, 4, 1]
(247) n=4096 Answer=[0, 0, 0, 12]
(248) n=4116 Answer=[3, 0, 1, 2]
(249) n=4200 Answer=[1, 2, 1, 3]
(250) n=4320 Answer=[0, 1, 3, 5]
(251) n=4374 Answer=[0, 0, 7, 1]
(252) n=4375 Answer=[1, 4, 0, 0]
(253) n=4410 Answer=[2, 1, 2, 1]
(254) n=4480 Answer=[1, 1, 0, 7]
(255) n=4500 Answer=[0, 3, 2, 2]
(256) n=4536 Answer=[1, 0, 4, 3]
(257) n=4608 Answer=[0, 0, 2, 9]
(258) n=4704 Answer=[2, 0, 1, 5]
(259) n=4725 Answer=[1, 2, 3, 0]
(260) n=4800 Answer=[0, 2, 1, 6]
(261) n=4802 Answer=[4, 0, 0, 1]
(262) n=4860 Answer=[0, 1, 5, 2]
(263) n=4900 Answer=[2, 2, 0, 2]
(264) n=5000 Answer=[0, 4, 0, 3]
(265) n=5040 Answer=[1, 1, 2, 4]
(266) n=5103 Answer=[1, 0, 6, 0]
(267) n=5120 Answer=[0, 1, 0, 10]
(268) n=5145 Answer=[3, 1, 1, 0]
(269) n=5184 Answer=[0, 0, 4, 6]
(270) n=5250 Answer=[1, 3, 1, 1]
(271) n=5292 Answer=[2, 0, 3, 2]
(272) n=5376 Answer=[1, 0, 1, 8]
(273) n=5400 Answer=[0, 2, 3, 3]
(274) n=5488 Answer=[3, 0, 0, 4]
(275) n=5600 Answer=[1, 2, 0, 5]
(276) n=5625 Answer=[0, 4, 2, 0]
(277) n=5670 Answer=[1, 1, 4, 1]
(278) n=5760 Answer=[0, 1, 2, 7]
(279) n=5832 Answer=[0, 0, 6, 3]
(280) n=5880 Answer=[2, 1, 1, 3]
(281) n=6000 Answer=[0, 3, 1, 4]
(282) n=6048 Answer=[1, 0, 3, 5]
(283) n=6075 Answer=[0, 2, 5, 0]
(284) n=6125 Answer=[2, 3, 0, 0]
(285) n=6144 Answer=[0, 0, 1, 11]
(286) n=6174 Answer=[3, 0, 2, 1]
(287) n=6250 Answer=[0, 5, 0, 1]
(288) n=6272 Answer=[2, 0, 0, 7]
(289) n=6300 Answer=[1, 2, 2, 2]
(290) n=6400 Answer=[0, 2, 0, 8]
(291) n=6480 Answer=[0, 1, 4, 4]
(292) n=6561 Answer=[0, 0, 8, 0]
(293) n=6615 Answer=[2, 1, 3, 0]
(294) n=6720 Answer=[1, 1, 1, 6]
(295) n=6750 Answer=[0, 3, 3, 1]
(296) n=6804 Answer=[1, 0, 5, 2]
(297) n=6860 Answer=[3, 1, 0, 2]
(298) n=6912 Answer=[0, 0, 3, 8]
(299) n=7000 Answer=[1, 3, 0, 3]
(300) n=7056 Answer=[2, 0, 2, 4]
(301) n=7168 Answer=[1, 0, 0, 10]
(302) n=7200 Answer=[0, 2, 2, 5]
(303) n=7203 Answer=[4, 0, 1, 0]
(304) n=7290 Answer=[0, 1, 6, 1]
(305) n=7350 Answer=[2, 2, 1, 1]
(306) n=7500 Answer=[0, 4, 1, 2]
(307) n=7560 Answer=[1, 1, 3, 3]
(308) n=7680 Answer=[0, 1, 1, 9]
(309) n=7776 Answer=[0, 0, 5, 5]
(310) n=7840 Answer=[2, 1, 0, 5]
(311) n=7875 Answer=[1, 3, 2, 0]
(312) n=7938 Answer=[2, 0, 4, 1]
(313) n=8000 Answer=[0, 3, 0, 6]
(314) n=8064 Answer=[1, 0, 2, 7]
(315) n=8100 Answer=[0, 2, 4, 2]
(316) n=8192 Answer=[0, 0, 0, 13]
(317) n=8232 Answer=[3, 0, 1, 3]
(318) n=8400 Answer=[1, 2, 1, 4]
(319) n=8505 Answer=[1, 1, 5, 0]
(320) n=8575 Answer=[3, 2, 0, 0]
(321) n=8640 Answer=[0, 1, 3, 6]
(322) n=8748 Answer=[0, 0, 7, 2]
(323) n=8750 Answer=[1, 4, 0, 1]
(324) n=8820 Answer=[2, 1, 2, 2]
(325) n=8960 Answer=[1, 1, 0, 8]
(326) n=9000 Answer=[0, 3, 2, 3]
(327) n=9072 Answer=[1, 0, 4, 4]
(328) n=9216 Answer=[0, 0, 2, 10]
(329) n=9261 Answer=[3, 0, 3, 0]
(330) n=9375 Answer=[0, 5, 1, 0]
(331) n=9408 Answer=[2, 0, 1, 6]
(332) n=9450 Answer=[1, 2, 3, 1]
(333) n=9600 Answer=[0, 2, 1, 7]
(334) n=9604 Answer=[4, 0, 0, 2]
(335) n=9720 Answer=[0, 1, 5, 3]
(336) n=9800 Answer=[2, 2, 0, 3]
(337) n=10000 Answer=[0, 4, 0, 4]
(338) n=10080 Answer=[1, 1, 2, 5]
(339) n=10125 Answer=[0, 3, 4, 0]
(340) n=10206 Answer=[1, 0, 6, 1]
(341) n=10240 Answer=[0, 1, 0, 11]
(342) n=10290 Answer=[3, 1, 1, 1]
(343) n=10368 Answer=[0, 0, 4, 7]
(344) n=10500 Answer=[1, 3, 1, 2]
(345) n=10584 Answer=[2, 0, 3, 3]
(346) n=10752 Answer=[1, 0, 1, 9]
(347) n=10800 Answer=[0, 2, 3, 4]
(348) n=10935 Answer=[0, 1, 7, 0]
(349) n=10976 Answer=[3, 0, 0, 5]
(350) n=11025 Answer=[2, 2, 2, 0]
(351) n=11200 Answer=[1, 2, 0, 6]
(352) n=11250 Answer=[0, 4, 2, 1]
(353) n=11340 Answer=[1, 1, 4, 2]
(354) n=11520 Answer=[0, 1, 2, 8]
(355) n=11664 Answer=[0, 0, 6, 4]
(356) n=11760 Answer=[2, 1, 1, 4]
(357) n=11907 Answer=[2, 0, 5, 0]
(358) n=12000 Answer=[0, 3, 1, 5]
(359) n=12005 Answer=[4, 1, 0, 0]
(360) n=12096 Answer=[1, 0, 3, 6]
(361) n=12150 Answer=[0, 2, 5, 1]
(362) n=12250 Answer=[2, 3, 0, 1]
(363) n=12288 Answer=[0, 0, 1, 12]
(364) n=12348 Answer=[3, 0, 2, 2]
(365) n=12500 Answer=[0, 5, 0, 2]
(366) n=12544 Answer=[2, 0, 0, 8]
(367) n=12600 Answer=[1, 2, 2, 3]
(368) n=12800 Answer=[0, 2, 0, 9]
(369) n=12960 Answer=[0, 1, 4, 5]
(370) n=13122 Answer=[0, 0, 8, 1]
(371) n=13125 Answer=[1, 4, 1, 0]
(372) n=13230 Answer=[2, 1, 3, 1]
(373) n=13440 Answer=[1, 1, 1, 7]
(374) n=13500 Answer=[0, 3, 3, 2]
(375) n=13608 Answer=[1, 0, 5, 3]
(376) n=13720 Answer=[3, 1, 0, 3]
(377) n=13824 Answer=[0, 0, 3, 9]
(378) n=14000 Answer=[1, 3, 0, 4]
(379) n=14112 Answer=[2, 0, 2, 5]
(380) n=14175 Answer=[1, 2, 4, 0]
(381) n=14336 Answer=[1, 0, 0, 11]
(382) n=14400 Answer=[0, 2, 2, 6]
(383) n=14406 Answer=[4, 0, 1, 1]
(384) n=14580 Answer=[0, 1, 6, 2]
(385) n=14700 Answer=[2, 2, 1, 2]
(386) n=15000 Answer=[0, 4, 1, 3]
(387) n=15120 Answer=[1, 1, 3, 4]
(388) n=15309 Answer=[1, 0, 7, 0]
(389) n=15360 Answer=[0, 1, 1, 10]
(390) n=15435 Answer=[3, 1, 2, 0]
(391) n=15552 Answer=[0, 0, 5, 6]
(392) n=15625 Answer=[0, 6, 0, 0]
(393) n=15680 Answer=[2, 1, 0, 6]
(394) n=15750 Answer=[1, 3, 2, 1]
(395) n=15876 Answer=[2, 0, 4, 2]
(396) n=16000 Answer=[0, 3, 0, 7]
(397) n=16128 Answer=[1, 0, 2, 8]
(398) n=16200 Answer=[0, 2, 4, 3]
(399) n=16384 Answer=[0, 0, 0, 14]
(400) n=16464 Answer=[3, 0, 1, 4]
(401) n=16800 Answer=[1, 2, 1, 5]
(402) n=16807 Answer=[5, 0, 0, 0]
(403) n=16875 Answer=[0, 4, 3, 0]
(404) n=17010 Answer=[1, 1, 5, 1]
(405) n=17150 Answer=[3, 2, 0, 1]
(406) n=17280 Answer=[0, 1, 3, 7]
(407) n=17496 Answer=[0, 0, 7, 3]
(408) n=17500 Answer=[1, 4, 0, 2]
(409) n=17640 Answer=[2, 1, 2, 3]
(410) n=17920 Answer=[1, 1, 0, 9]
(411) n=18000 Answer=[0, 3, 2, 4]
(412) n=18144 Answer=[1, 0, 4, 5]
(413) n=18225 Answer=[0, 2, 6, 0]
(414) n=18375 Answer=[2, 3, 1, 0]
(415) n=18432 Answer=[0, 0, 2, 11]
(416) n=18522 Answer=[3, 0, 3, 1]
(417) n=18750 Answer=[0, 5, 1, 1]
(418) n=18816 Answer=[2, 0, 1, 7]
(419) n=18900 Answer=[1, 2, 3, 2]
(420) n=19200 Answer=[0, 2, 1, 8]
(421) n=19208 Answer=[4, 0, 0, 3]
(422) n=19440 Answer=[0, 1, 5, 4]
(423) n=19600 Answer=[2, 2, 0, 4]
(424) n=19683 Answer=[0, 0, 9, 0]
(425) n=19845 Answer=[2, 1, 4, 0]
(426) n=20000 Answer=[0, 4, 0, 5]
(427) n=20160 Answer=[1, 1, 2, 6]
(428) n=20250 Answer=[0, 3, 4, 1]
(429) n=20412 Answer=[1, 0, 6, 2]
(430) n=20480 Answer=[0, 1, 0, 12]
(431) n=20580 Answer=[3, 1, 1, 2]
(432) n=20736 Answer=[0, 0, 4, 8]
(433) n=21000 Answer=[1, 3, 1, 3]
(434) n=21168 Answer=[2, 0, 3, 4]
(435) n=21504 Answer=[1, 0, 1, 10]
(436) n=21600 Answer=[0, 2, 3, 5]
(437) n=21609 Answer=[4, 0, 2, 0]
(438) n=21870 Answer=[0, 1, 7, 1]
(439) n=21875 Answer=[1, 5, 0, 0]
(440) n=21952 Answer=[3, 0, 0, 6]
(441) n=22050 Answer=[2, 2, 2, 1]
(442) n=22400 Answer=[1, 2, 0, 7]
(443) n=22500 Answer=[0, 4, 2, 2]
(444) n=22680 Answer=[1, 1, 4, 3]
(445) n=23040 Answer=[0, 1, 2, 9]
(446) n=23328 Answer=[0, 0, 6, 5]
(447) n=23520 Answer=[2, 1, 1, 5]
(448) n=23625 Answer=[1, 3, 3, 0]
(449) n=23814 Answer=[2, 0, 5, 1]
(450) n=24000 Answer=[0, 3, 1, 6]
(451) n=24010 Answer=[4, 1, 0, 1]
(452) n=24192 Answer=[1, 0, 3, 7]
(453) n=24300 Answer=[0, 2, 5, 2]
(454) n=24500 Answer=[2, 3, 0, 2]
(455) n=24576 Answer=[0, 0, 1, 13]
(456) n=24696 Answer=[3, 0, 2, 3]
(457) n=25000 Answer=[0, 5, 0, 3]
(458) n=25088 Answer=[2, 0, 0, 9]
(459) n=25200 Answer=[1, 2, 2, 4]
(460) n=25515 Answer=[1, 1, 6, 0]
(461) n=25600 Answer=[0, 2, 0, 10]
(462) n=25725 Answer=[3, 2, 1, 0]
(463) n=25920 Answer=[0, 1, 4, 6]
(464) n=26244 Answer=[0, 0, 8, 2]
(465) n=26250 Answer=[1, 4, 1, 1]
(466) n=26460 Answer=[2, 1, 3, 2]
(467) n=26880 Answer=[1, 1, 1, 8]
(468) n=27000 Answer=[0, 3, 3, 3]
(469) n=27216 Answer=[1, 0, 5, 4]
(470) n=27440 Answer=[3, 1, 0, 4]
(471) n=27648 Answer=[0, 0, 3, 10]
(472) n=27783 Answer=[3, 0, 4, 0]
(473) n=28000 Answer=[1, 3, 0, 5]
(474) n=28125 Answer=[0, 5, 2, 0]
(475) n=28224 Answer=[2, 0, 2, 6]
(476) n=28350 Answer=[1, 2, 4, 1]
(477) n=28672 Answer=[1, 0, 0, 12]
(478) n=28800 Answer=[0, 2, 2, 7]
(479) n=28812 Answer=[4, 0, 1, 2]
(480) n=29160 Answer=[0, 1, 6, 3]
(481) n=29400 Answer=[2, 2, 1, 3]
(482) n=30000 Answer=[0, 4, 1, 4]
(483) n=30240 Answer=[1, 1, 3, 5]
(484) n=30375 Answer=[0, 3, 5, 0]
(485) n=30618 Answer=[1, 0, 7, 1]
(486) n=30625 Answer=[2, 4, 0, 0]
(487) n=30720 Answer=[0, 1, 1, 11]
(488) n=30870 Answer=[3, 1, 2, 1]
(489) n=31104 Answer=[0, 0, 5, 7]
(490) n=31250 Answer=[0, 6, 0, 1]
(491) n=31360 Answer=[2, 1, 0, 7]
(492) n=31500 Answer=[1, 3, 2, 2]
(493) n=31752 Answer=[2, 0, 4, 3]
(494) n=32000 Answer=[0, 3, 0, 8]
(495) n=32256 Answer=[1, 0, 2, 9]
(496) n=32400 Answer=[0, 2, 4, 4]
(497) n=32768 Answer=[0, 0, 0, 15]
(498) n=32805 Answer=[0, 1, 8, 0]
(499) n=32928 Answer=[3, 0, 1, 5]
(500) n=33075 Answer=[2, 2, 3, 0]
(501) n=33600 Answer=[1, 2, 1, 6]
(502) n=33614 Answer=[5, 0, 0, 1]
(503) n=33750 Answer=[0, 4, 3, 1]
(504) n=34020 Answer=[1, 1, 5, 2]
(505) n=34300 Answer=[3, 2, 0, 2]
(506) n=34560 Answer=[0, 1, 3, 8]
(507) n=34992 Answer=[0, 0, 7, 4]
(508) n=35000 Answer=[1, 4, 0, 3]
(509) n=35280 Answer=[2, 1, 2, 4]
(510) n=35721 Answer=[2, 0, 6, 0]
(511) n=35840 Answer=[1, 1, 0, 10]
(512) n=36000 Answer=[0, 3, 2, 5]
(513) n=36015 Answer=[4, 1, 1, 0]
(514) n=36288 Answer=[1, 0, 4, 6]
(515) n=36450 Answer=[0, 2, 6, 1]
(516) n=36750 Answer=[2, 3, 1, 1]
(517) n=36864 Answer=[0, 0, 2, 12]
(518) n=37044 Answer=[3, 0, 3, 2]
(519) n=37500 Answer=[0, 5, 1, 2]
(520) n=37632 Answer=[2, 0, 1, 8]
(521) n=37800 Answer=[1, 2, 3, 3]
(522) n=38400 Answer=[0, 2, 1, 9]
(523) n=38416 Answer=[4, 0, 0, 4]
(524) n=38880 Answer=[0, 1, 5, 5]
(525) n=39200 Answer=[2, 2, 0, 5]
(526) n=39366 Answer=[0, 0, 9, 1]
(527) n=39375 Answer=[1, 4, 2, 0]
(528) n=39690 Answer=[2, 1, 4, 1]
(529) n=40000 Answer=[0, 4, 0, 6]
(530) n=40320 Answer=[1, 1, 2, 7]
(531) n=40500 Answer=[0, 3, 4, 2]
(532) n=40824 Answer=[1, 0, 6, 3]
(533) n=40960 Answer=[0, 1, 0, 13]
(534) n=41160 Answer=[3, 1, 1, 3]
(535) n=41472 Answer=[0, 0, 4, 9]
(536) n=42000 Answer=[1, 3, 1, 4]
(537) n=42336 Answer=[2, 0, 3, 5]
(538) n=42525 Answer=[1, 2, 5, 0]
(539) n=42875 Answer=[3, 3, 0, 0]
(540) n=43008 Answer=[1, 0, 1, 11]
(541) n=43200 Answer=[0, 2, 3, 6]
(542) n=43218 Answer=[4, 0, 2, 1]
(543) n=43740 Answer=[0, 1, 7, 2]
(544) n=43750 Answer=[1, 5, 0, 1]
(545) n=43904 Answer=[3, 0, 0, 7]
(546) n=44100 Answer=[2, 2, 2, 2]
(547) n=44800 Answer=[1, 2, 0, 8]
(548) n=45000 Answer=[0, 4, 2, 3]
(549) n=45360 Answer=[1, 1, 4, 4]
(550) n=45927 Answer=[1, 0, 8, 0]
(551) n=46080 Answer=[0, 1, 2, 10]
(552) n=46305 Answer=[3, 1, 3, 0]
(553) n=46656 Answer=[0, 0, 6, 6]
(554) n=46875 Answer=[0, 6, 1, 0]
(555) n=47040 Answer=[2, 1, 1, 6]
(556) n=47250 Answer=[1, 3, 3, 1]
(557) n=47628 Answer=[2, 0, 5, 2]
(558) n=48000 Answer=[0, 3, 1, 7]
(559) n=48020 Answer=[4, 1, 0, 2]
(560) n=48384 Answer=[1, 0, 3, 8]
(561) n=48600 Answer=[0, 2, 5, 3]
(562) n=49000 Answer=[2, 3, 0, 3]
(563) n=49152 Answer=[0, 0, 1, 14]
(564) n=49392 Answer=[3, 0, 2, 4]
(565) n=50000 Answer=[0, 5, 0, 4]
(566) n=50176 Answer=[2, 0, 0, 10]
(567) n=50400 Answer=[1, 2, 2, 5]
(568) n=50421 Answer=[5, 0, 1, 0]
(569) n=50625 Answer=[0, 4, 4, 0]
(570) n=51030 Answer=[1, 1, 6, 1]
(571) n=51200 Answer=[0, 2, 0, 11]
(572) n=51450 Answer=[3, 2, 1, 1]
(573) n=51840 Answer=[0, 1, 4, 7]
(574) n=52488 Answer=[0, 0, 8, 3]
(575) n=52500 Answer=[1, 4, 1, 2]
(576) n=52920 Answer=[2, 1, 3, 3]
(577) n=53760 Answer=[1, 1, 1, 9]
(578) n=54000 Answer=[0, 3, 3, 4]
(579) n=54432 Answer=[1, 0, 5, 5]
(580) n=54675 Answer=[0, 2, 7, 0]
(581) n=54880 Answer=[3, 1, 0, 5]
(582) n=55125 Answer=[2, 3, 2, 0]
(583) n=55296 Answer=[0, 0, 3, 11]
(584) n=55566 Answer=[3, 0, 4, 1]
(585) n=56000 Answer=[1, 3, 0, 6]
(586) n=56250 Answer=[0, 5, 2, 1]
(587) n=56448 Answer=[2, 0, 2, 7]
(588) n=56700 Answer=[1, 2, 4, 2]
(589) n=57344 Answer=[1, 0, 0, 13]
(590) n=57600 Answer=[0, 2, 2, 8]
(591) n=57624 Answer=[4, 0, 1, 3]
(592) n=58320 Answer=[0, 1, 6, 4]
(593) n=58800 Answer=[2, 2, 1, 4]
(594) n=59049 Answer=[0, 0, 10, 0]
(595) n=59535 Answer=[2, 1, 5, 0]
(596) n=60000 Answer=[0, 4, 1, 5]
(597) n=60025 Answer=[4, 2, 0, 0]
(598) n=60480 Answer=[1, 1, 3, 6]
(599) n=60750 Answer=[0, 3, 5, 1]
(600) n=61236 Answer=[1, 0, 7, 2]
(601) n=61250 Answer=[2, 4, 0, 1]
(602) n=61440 Answer=[0, 1, 1, 12]
(603) n=61740 Answer=[3, 1, 2, 2]
(604) n=62208 Answer=[0, 0, 5, 8]
(605) n=62500 Answer=[0, 6, 0, 2]
(606) n=62720 Answer=[2, 1, 0, 8]
(607) n=63000 Answer=[1, 3, 2, 3]
(608) n=63504 Answer=[2, 0, 4, 4]
(609) n=64000 Answer=[0, 3, 0, 9]
(610) n=64512 Answer=[1, 0, 2, 10]
(611) n=64800 Answer=[0, 2, 4, 5]
(612) n=64827 Answer=[4, 0, 3, 0]
(613) n=65536 Answer=[0, 0, 0, 16]
(614) n=65610 Answer=[0, 1, 8, 1]
(615) n=65625 Answer=[1, 5, 1, 0]
(616) n=65856 Answer=[3, 0, 1, 6]
(617) n=66150 Answer=[2, 2, 3, 1]
(618) n=67200 Answer=[1, 2, 1, 7]
(619) n=67228 Answer=[5, 0, 0, 2]
(620) n=67500 Answer=[0, 4, 3, 2]
(621) n=68040 Answer=[1, 1, 5, 3]
(622) n=68600 Answer=[3, 2, 0, 3]
(623) n=69120 Answer=[0, 1, 3, 9]
(624) n=69984 Answer=[0, 0, 7, 5]
(625) n=70000 Answer=[1, 4, 0, 4]
(626) n=70560 Answer=[2, 1, 2, 5]
(627) n=70875 Answer=[1, 3, 4, 0]
(628) n=71442 Answer=[2, 0, 6, 1]
(629) n=71680 Answer=[1, 1, 0, 11]
(630) n=72000 Answer=[0, 3, 2, 6]
(631) n=72030 Answer=[4, 1, 1, 1]
(632) n=72576 Answer=[1, 0, 4, 7]
(633) n=72900 Answer=[0, 2, 6, 2]
(634) n=73500 Answer=[2, 3, 1, 2]
(635) n=73728 Answer=[0, 0, 2, 13]
(636) n=74088 Answer=[3, 0, 3, 3]
(637) n=75000 Answer=[0, 5, 1, 3]
(638) n=75264 Answer=[2, 0, 1, 9]
(639) n=75600 Answer=[1, 2, 3, 4]
(640) n=76545 Answer=[1, 1, 7, 0]
(641) n=76800 Answer=[0, 2, 1, 10]
(642) n=76832 Answer=[4, 0, 0, 5]
(643) n=77175 Answer=[3, 2, 2, 0]
(644) n=77760 Answer=[0, 1, 5, 6]
(645) n=78125 Answer=[0, 7, 0, 0]
(646) n=78400 Answer=[2, 2, 0, 6]
(647) n=78732 Answer=[0, 0, 9, 2]
(648) n=78750 Answer=[1, 4, 2, 1]
(649) n=79380 Answer=[2, 1, 4, 2]
(650) n=80000 Answer=[0, 4, 0, 7]
(651) n=80640 Answer=[1, 1, 2, 8]
(652) n=81000 Answer=[0, 3, 4, 3]
(653) n=81648 Answer=[1, 0, 6, 4]
(654) n=81920 Answer=[0, 1, 0, 14]
(655) n=82320 Answer=[3, 1, 1, 4]
(656) n=82944 Answer=[0, 0, 4, 10]
(657) n=83349 Answer=[3, 0, 5, 0]
(658) n=84000 Answer=[1, 3, 1, 5]
(659) n=84035 Answer=[5, 1, 0, 0]
(660) n=84375 Answer=[0, 5, 3, 0]
(661) n=84672 Answer=[2, 0, 3, 6]
(662) n=85050 Answer=[1, 2, 5, 1]
(663) n=85750 Answer=[3, 3, 0, 1]
(664) n=86016 Answer=[1, 0, 1, 12]
(665) n=86400 Answer=[0, 2, 3, 7]
(666) n=86436 Answer=[4, 0, 2, 2]
(667) n=87480 Answer=[0, 1, 7, 3]
(668) n=87500 Answer=[1, 5, 0, 2]
(669) n=87808 Answer=[3, 0, 0, 8]
(670) n=88200 Answer=[2, 2, 2, 3]
(671) n=89600 Answer=[1, 2, 0, 9]
(672) n=90000 Answer=[0, 4, 2, 4]
(673) n=90720 Answer=[1, 1, 4, 5]
(674) n=91125 Answer=[0, 3, 6, 0]
(675) n=91854 Answer=[1, 0, 8, 1]
(676) n=91875 Answer=[2, 4, 1, 0]
(677) n=92160 Answer=[0, 1, 2, 11]
(678) n=92610 Answer=[3, 1, 3, 1]
(679) n=93312 Answer=[0, 0, 6, 7]
(680) n=93750 Answer=[0, 6, 1, 1]
(681) n=94080 Answer=[2, 1, 1, 7]
(682) n=94500 Answer=[1, 3, 3, 2]
(683) n=95256 Answer=[2, 0, 5, 3]
(684) n=96000 Answer=[0, 3, 1, 8]
(685) n=96040 Answer=[4, 1, 0, 3]
(686) n=96768 Answer=[1, 0, 3, 9]
(687) n=97200 Answer=[0, 2, 5, 4]
(688) n=98000 Answer=[2, 3, 0, 4]
(689) n=98304 Answer=[0, 0, 1, 15]
(690) n=98415 Answer=[0, 1, 9, 0]
(691) n=98784 Answer=[3, 0, 2, 5]
(692) n=99225 Answer=[2, 2, 4, 0]
(693) n=100000 Answer=[0, 5, 0, 5]
(694) n=100352 Answer=[2, 0, 0, 11]
(695) n=100800 Answer=[1, 2, 2, 6]
(696) n=100842 Answer=[5, 0, 1, 1]
(697) n=101250 Answer=[0, 4, 4, 1]
(698) n=102060 Answer=[1, 1, 6, 2]
(699) n=102400 Answer=[0, 2, 0, 12]
(700) n=102900 Answer=[3, 2, 1, 2]
(701) n=103680 Answer=[0, 1, 4, 8]
(702) n=104976 Answer=[0, 0, 8, 4]
(703) n=105000 Answer=[1, 4, 1, 3]
(704) n=105840 Answer=[2, 1, 3, 4]
(705) n=107163 Answer=[2, 0, 7, 0]
(706) n=107520 Answer=[1, 1, 1, 10]
(707) n=108000 Answer=[0, 3, 3, 5]
(708) n=108045 Answer=[4, 1, 2, 0]
(709) n=108864 Answer=[1, 0, 5, 6]
(710) n=109350 Answer=[0, 2, 7, 1]
(711) n=109375 Answer=[1, 6, 0, 0]
(712) n=109760 Answer=[3, 1, 0, 6]
(713) n=110250 Answer=[2, 3, 2, 1]
(714) n=110592 Answer=[0, 0, 3, 12]
(715) n=111132 Answer=[3, 0, 4, 2]
(716) n=112000 Answer=[1, 3, 0, 7]
(717) n=112500 Answer=[0, 5, 2, 2]
(718) n=112896 Answer=[2, 0, 2, 8]
(719) n=113400 Answer=[1, 2, 4, 3]
(720) n=114688 Answer=[1, 0, 0, 14]
(721) n=115200 Answer=[0, 2, 2, 9]
(722) n=115248 Answer=[4, 0, 1, 4]
(723) n=116640 Answer=[0, 1, 6, 5]
(724) n=117600 Answer=[2, 2, 1, 5]
(725) n=117649 Answer=[6, 0, 0, 0]
(726) n=118098 Answer=[0, 0, 10, 1]
(727) n=118125 Answer=[1, 4, 3, 0]
(728) n=119070 Answer=[2, 1, 5, 1]
(729) n=120000 Answer=[0, 4, 1, 6]
(730) n=120050 Answer=[4, 2, 0, 1]
(731) n=120960 Answer=[1, 1, 3, 7]
(732) n=121500 Answer=[0, 3, 5, 2]
(733) n=122472 Answer=[1, 0, 7, 3]
(734) n=122500 Answer=[2, 4, 0, 2]
(735) n=122880 Answer=[0, 1, 1, 13]
(736) n=123480 Answer=[3, 1, 2, 3]
(737) n=124416 Answer=[0, 0, 5, 9]
(738) n=125000 Answer=[0, 6, 0, 3]
(739) n=125440 Answer=[2, 1, 0, 9]
(740) n=126000 Answer=[1, 3, 2, 4]
(741) n=127008 Answer=[2, 0, 4, 5]
(742) n=127575 Answer=[1, 2, 6, 0]
(743) n=128000 Answer=[0, 3, 0, 10]
(744) n=128625 Answer=[3, 3, 1, 0]
(745) n=129024 Answer=[1, 0, 2, 11]
(746) n=129600 Answer=[0, 2, 4, 6]
(747) n=129654 Answer=[4, 0, 3, 1]
(748) n=131072 Answer=[0, 0, 0, 17]
(749) n=131220 Answer=[0, 1, 8, 2]
(750) n=131250 Answer=[1, 5, 1, 1]
(751) n=131712 Answer=[3, 0, 1, 7]
(752) n=132300 Answer=[2, 2, 3, 2]
(753) n=134400 Answer=[1, 2, 1, 8]
(754) n=134456 Answer=[5, 0, 0, 3]
(755) n=135000 Answer=[0, 4, 3, 3]
(756) n=136080 Answer=[1, 1, 5, 4]
(757) n=137200 Answer=[3, 2, 0, 4]
(758) n=137781 Answer=[1, 0, 9, 0]
(759) n=138240 Answer=[0, 1, 3, 10]
(760) n=138915 Answer=[3, 1, 4, 0]
(761) n=139968 Answer=[0, 0, 7, 6]
(762) n=140000 Answer=[1, 4, 0, 5]
(763) n=140625 Answer=[0, 6, 2, 0]
(764) n=141120 Answer=[2, 1, 2, 6]
(765) n=141750 Answer=[1, 3, 4, 1]
(766) n=142884 Answer=[2, 0, 6, 2]
(767) n=143360 Answer=[1, 1, 0, 12]
(768) n=144000 Answer=[0, 3, 2, 7]
(769) n=144060 Answer=[4, 1, 1, 2]
(770) n=145152 Answer=[1, 0, 4, 8]
(771) n=145800 Answer=[0, 2, 6, 3]
(772) n=147000 Answer=[2, 3, 1, 3]
(773) n=147456 Answer=[0, 0, 2, 14]
(774) n=148176 Answer=[3, 0, 3, 4]
(775) n=150000 Answer=[0, 5, 1, 4]
(776) n=150528 Answer=[2, 0, 1, 10]
(777) n=151200 Answer=[1, 2, 3, 5]
(778) n=151263 Answer=[5, 0, 2, 0]
(779) n=151875 Answer=[0, 4, 5, 0]
(780) n=153090 Answer=[1, 1, 7, 1]
(781) n=153125 Answer=[2, 5, 0, 0]
(782) n=153600 Answer=[0, 2, 1, 11]
(783) n=153664 Answer=[4, 0, 0, 6]
(784) n=154350 Answer=[3, 2, 2, 1]
(785) n=155520 Answer=[0, 1, 5, 7]
(786) n=156250 Answer=[0, 7, 0, 1]
(787) n=156800 Answer=[2, 2, 0, 7]
(788) n=157464 Answer=[0, 0, 9, 3]
(789) n=157500 Answer=[1, 4, 2, 2]
(790) n=158760 Answer=[2, 1, 4, 3]
(791) n=160000 Answer=[0, 4, 0, 8]
(792) n=161280 Answer=[1, 1, 2, 9]
(793) n=162000 Answer=[0, 3, 4, 4]
(794) n=163296 Answer=[1, 0, 6, 5]
(795) n=163840 Answer=[0, 1, 0, 15]
(796) n=164025 Answer=[0, 2, 8, 0]
(797) n=164640 Answer=[3, 1, 1, 5]
(798) n=165375 Answer=[2, 3, 3, 0]
(799) n=165888 Answer=[0, 0, 4, 11]
(800) n=166698 Answer=[3, 0, 5, 1]
(801) n=168000 Answer=[1, 3, 1, 6]
(802) n=168070 Answer=[5, 1, 0, 1]
(803) n=168750 Answer=[0, 5, 3, 1]
(804) n=169344 Answer=[2, 0, 3, 7]
(805) n=170100 Answer=[1, 2, 5, 2]
(806) n=171500 Answer=[3, 3, 0, 2]
(807) n=172032 Answer=[1, 0, 1, 13]
(808) n=172800 Answer=[0, 2, 3, 8]
(809) n=172872 Answer=[4, 0, 2, 3]
(810) n=174960 Answer=[0, 1, 7, 4]
(811) n=175000 Answer=[1, 5, 0, 3]
(812) n=175616 Answer=[3, 0, 0, 9]
(813) n=176400 Answer=[2, 2, 2, 4]
(814) n=177147 Answer=[0, 0, 11, 0]
(815) n=178605 Answer=[2, 1, 6, 0]
(816) n=179200 Answer=[1, 2, 0, 10]
(817) n=180000 Answer=[0, 4, 2, 5]
(818) n=180075 Answer=[4, 2, 1, 0]
(819) n=181440 Answer=[1, 1, 4, 6]
(820) n=182250 Answer=[0, 3, 6, 1]
(821) n=183708 Answer=[1, 0, 8, 2]
(822) n=183750 Answer=[2, 4, 1, 1]
(823) n=184320 Answer=[0, 1, 2, 12]
(824) n=185220 Answer=[3, 1, 3, 2]
(825) n=186624 Answer=[0, 0, 6, 8]
(826) n=187500 Answer=[0, 6, 1, 2]
(827) n=188160 Answer=[2, 1, 1, 8]
(828) n=189000 Answer=[1, 3, 3, 3]
(829) n=190512 Answer=[2, 0, 5, 4]
(830) n=192000 Answer=[0, 3, 1, 9]
(831) n=192080 Answer=[4, 1, 0, 4]
(832) n=193536 Answer=[1, 0, 3, 10]
(833) n=194400 Answer=[0, 2, 5, 5]
(834) n=194481 Answer=[4, 0, 4, 0]
(835) n=196000 Answer=[2, 3, 0, 5]
(836) n=196608 Answer=[0, 0, 1, 16]
(837) n=196830 Answer=[0, 1, 9, 1]
(838) n=196875 Answer=[1, 5, 2, 0]
(839) n=197568 Answer=[3, 0, 2, 6]
(840) n=198450 Answer=[2, 2, 4, 1]
(841) n=200000 Answer=[0, 5, 0, 6]
(842) n=200704 Answer=[2, 0, 0, 12]
(843) n=201600 Answer=[1, 2, 2, 7]
(844) n=201684 Answer=[5, 0, 1, 2]
(845) n=202500 Answer=[0, 4, 4, 2]
(846) n=204120 Answer=[1, 1, 6, 3]
(847) n=204800 Answer=[0, 2, 0, 13]
(848) n=205800 Answer=[3, 2, 1, 3]
(849) n=207360 Answer=[0, 1, 4, 9]
(850) n=209952 Answer=[0, 0, 8, 5]
(851) n=210000 Answer=[1, 4, 1, 4]
(852) n=211680 Answer=[2, 1, 3, 5]
(853) n=212625 Answer=[1, 3, 5, 0]
(854) n=214326 Answer=[2, 0, 7, 1]
(855) n=214375 Answer=[3, 4, 0, 0]
(856) n=215040 Answer=[1, 1, 1, 11]
(857) n=216000 Answer=[0, 3, 3, 6]
(858) n=216090 Answer=[4, 1, 2, 1]
(859) n=217728 Answer=[1, 0, 5, 7]
(860) n=218700 Answer=[0, 2, 7, 2]
(861) n=218750 Answer=[1, 6, 0, 1]
(862) n=219520 Answer=[3, 1, 0, 7]
(863) n=220500 Answer=[2, 3, 2, 2]
(864) n=221184 Answer=[0, 0, 3, 13]
(865) n=222264 Answer=[3, 0, 4, 3]
(866) n=224000 Answer=[1, 3, 0, 8]
(867) n=225000 Answer=[0, 5, 2, 3]
(868) n=225792 Answer=[2, 0, 2, 9]
(869) n=226800 Answer=[1, 2, 4, 4]
(870) n=229376 Answer=[1, 0, 0, 15]
(871) n=229635 Answer=[1, 1, 8, 0]
(872) n=230400 Answer=[0, 2, 2, 10]
(873) n=230496 Answer=[4, 0, 1, 5]
(874) n=231525 Answer=[3, 2, 3, 0]
(875) n=233280 Answer=[0, 1, 6, 6]
(876) n=234375 Answer=[0, 7, 1, 0]
(877) n=235200 Answer=[2, 2, 1, 6]
(878) n=235298 Answer=[6, 0, 0, 1]
(879) n=236196 Answer=[0, 0, 10, 2]
(880) n=236250 Answer=[1, 4, 3, 1]
(881) n=238140 Answer=[2, 1, 5, 2]
(882) n=240000 Answer=[0, 4, 1, 7]
(883) n=240100 Answer=[4, 2, 0, 2]
(884) n=241920 Answer=[1, 1, 3, 8]
(885) n=243000 Answer=[0, 3, 5, 3]
(886) n=244944 Answer=[1, 0, 7, 4]
(887) n=245000 Answer=[2, 4, 0, 3]
(888) n=245760 Answer=[0, 1, 1, 14]
(889) n=246960 Answer=[3, 1, 2, 4]
(890) n=248832 Answer=[0, 0, 5, 10]
(891) n=250000 Answer=[0, 6, 0, 4]
(892) n=250047 Answer=[3, 0, 6, 0]
(893) n=250880 Answer=[2, 1, 0, 10]
(894) n=252000 Answer=[1, 3, 2, 5]
(895) n=252105 Answer=[5, 1, 1, 0]
(896) n=253125 Answer=[0, 5, 4, 0]
(897) n=254016 Answer=[2, 0, 4, 6]
(898) n=255150 Answer=[1, 2, 6, 1]
(899) n=256000 Answer=[0, 3, 0, 11]
(900) n=257250 Answer=[3, 3, 1, 1]
(901) n=258048 Answer=[1, 0, 2, 12]
(902) n=259200 Answer=[0, 2, 4, 7]
(903) n=259308 Answer=[4, 0, 3, 2]
(904) n=262144 Answer=[0, 0, 0, 18]
(905) n=262440 Answer=[0, 1, 8, 3]
(906) n=262500 Answer=[1, 5, 1, 2]
(907) n=263424 Answer=[3, 0, 1, 8]
(908) n=264600 Answer=[2, 2, 3, 3]
(909) n=268800 Answer=[1, 2, 1, 9]
(910) n=268912 Answer=[5, 0, 0, 4]
(911) n=270000 Answer=[0, 4, 3, 4]
(912) n=272160 Answer=[1, 1, 5, 5]
(913) n=273375 Answer=[0, 3, 7, 0]
(914) n=274400 Answer=[3, 2, 0, 5]
(915) n=275562 Answer=[1, 0, 9, 1]
(916) n=275625 Answer=[2, 4, 2, 0]
(917) n=276480 Answer=[0, 1, 3, 11]
(918) n=277830 Answer=[3, 1, 4, 1]
(919) n=279936 Answer=[0, 0, 7, 7]
(920) n=280000 Answer=[1, 4, 0, 6]
(921) n=281250 Answer=[0, 6, 2, 1]
(922) n=282240 Answer=[2, 1, 2, 7]
(923) n=283500 Answer=[1, 3, 4, 2]
(924) n=285768 Answer=[2, 0, 6, 3]
(925) n=286720 Answer=[1, 1, 0, 13]
(926) n=288000 Answer=[0, 3, 2, 8]
(927) n=288120 Answer=[4, 1, 1, 3]
(928) n=290304 Answer=[1, 0, 4, 9]
(929) n=291600 Answer=[0, 2, 6, 4]
(930) n=294000 Answer=[2, 3, 1, 4]
(931) n=294912 Answer=[0, 0, 2, 15]
(932) n=295245 Answer=[0, 1, 10, 0]
(933) n=296352 Answer=[3, 0, 3, 5]
(934) n=297675 Answer=[2, 2, 5, 0]
(935) n=300000 Answer=[0, 5, 1, 5]
(936) n=300125 Answer=[4, 3, 0, 0]
(937) n=301056 Answer=[2, 0, 1, 11]
(938) n=302400 Answer=[1, 2, 3, 6]
(939) n=302526 Answer=[5, 0, 2, 1]
(940) n=303750 Answer=[0, 4, 5, 1]
(941) n=306180 Answer=[1, 1, 7, 2]
(942) n=306250 Answer=[2, 5, 0, 1]
(943) n=307200 Answer=[0, 2, 1, 12]
(944) n=307328 Answer=[4, 0, 0, 7]
(945) n=308700 Answer=[3, 2, 2, 2]
(946) n=311040 Answer=[0, 1, 5, 8]
(947) n=312500 Answer=[0, 7, 0, 2]
(948) n=313600 Answer=[2, 2, 0, 8]
(949) n=314928 Answer=[0, 0, 9, 4]
(950) n=315000 Answer=[1, 4, 2, 3]
(951) n=317520 Answer=[2, 1, 4, 4]
(952) n=320000 Answer=[0, 4, 0, 9]
(953) n=321489 Answer=[2, 0, 8, 0]
(954) n=322560 Answer=[1, 1, 2, 10]
(955) n=324000 Answer=[0, 3, 4, 5]
(956) n=324135 Answer=[4, 1, 3, 0]
(957) n=326592 Answer=[1, 0, 6, 6]
(958) n=327680 Answer=[0, 1, 0, 16]
(959) n=328050 Answer=[0, 2, 8, 1]
(960) n=328125 Answer=[1, 6, 1, 0]
(961) n=329280 Answer=[3, 1, 1, 6]
(962) n=330750 Answer=[2, 3, 3, 1]
(963) n=331776 Answer=[0, 0, 4, 12]
(964) n=333396 Answer=[3, 0, 5, 2]
(965) n=336000 Answer=[1, 3, 1, 7]
(966) n=336140 Answer=[5, 1, 0, 2]
(967) n=337500 Answer=[0, 5, 3, 2]
(968) n=338688 Answer=[2, 0, 3, 8]
(969) n=340200 Answer=[1, 2, 5, 3]
(970) n=343000 Answer=[3, 3, 0, 3]
(971) n=344064 Answer=[1, 0, 1, 14]
(972) n=345600 Answer=[0, 2, 3, 9]
(973) n=345744 Answer=[4, 0, 2, 4]
(974) n=349920 Answer=[0, 1, 7, 5]
(975) n=350000 Answer=[1, 5, 0, 4]
(976) n=351232 Answer=[3, 0, 0, 10]
(977) n=352800 Answer=[2, 2, 2, 5]
(978) n=352947 Answer=[6, 0, 1, 0]
(979) n=354294 Answer=[0, 0, 11, 1]
(980) n=354375 Answer=[1, 4, 4, 0]
(981) n=357210 Answer=[2, 1, 6, 1]
(982) n=358400 Answer=[1, 2, 0, 11]
(983) n=360000 Answer=[0, 4, 2, 6]
(984) n=360150 Answer=[4, 2, 1, 1]
(985) n=362880 Answer=[1, 1, 4, 7]
(986) n=364500 Answer=[0, 3, 6, 2]
(987) n=367416 Answer=[1, 0, 8, 3]
(988) n=367500 Answer=[2, 4, 1, 2]
(989) n=368640 Answer=[0, 1, 2, 13]
(990) n=370440 Answer=[3, 1, 3, 3]
(991) n=373248 Answer=[0, 0, 6, 9]
(992) n=375000 Answer=[0, 6, 1, 3]
(993) n=376320 Answer=[2, 1, 1, 9]
(994) n=378000 Answer=[1, 3, 3, 4]
(995) n=381024 Answer=[2, 0, 5, 5]
(996) n=382725 Answer=[1, 2, 7, 0]
(997) n=384000 Answer=[0, 3, 1, 10]
(998) n=384160 Answer=[4, 1, 0, 5]
(999) n=385875 Answer=[3, 3, 2, 0]
(1000) n=387072 Answer=[1, 0, 3, 11]

「　👆　満足かだぜ、お父ん？」

「　現代のパソコンでも　計算は１秒では終わらないんだなあ」

(0) n=1 Answer=[0, 0, 0, 0]
(9) n=10 Answer=[0, 1, 0, 1]
(45) n=100 Answer=[0, 2, 0, 2]
(140) n=1000 Answer=[0, 3, 0, 3]
(337) n=10000 Answer=[0, 4, 0, 4]
(693) n=100000 Answer=[0, 5, 0, 5]

「　👆　１０倍の価値の現れ方が　面白いよな？」

「　指数の底（てい）が　２　と　５　ですからねえ」

対数を使って、アルゴリズムを高速化しようぜ？

「　n が 30万を超えてくると、割り切れる数を探すだけで　数秒かかってるしな……」

「　ベクトルの成分、 20 が出てくることも稀なのに　100　で回してるしな」

(904) n=262144 Answer=[0, 0, 0, 18]

「　👆　例えば　2 の 18乗は　262144　だが、
底（てい）を２とする 262144 の対数は 18 だぜ。 log2 262144 = 18」

「　ループする回数を、対数を使って　決めましょう」

# 素因数分解の練習
#
# cd math/prime_factorization
# python step1_1.py

import math

def factorize(n, debug=True):

    # int( ... ) だと、取りこぼしてしまうようだ。 math.ceil( ... ) とした
    # + 1 が必要
    a_max = math.ceil(math.log(n, 7))+1
    b_max = math.ceil(math.log(n, 5))+1
    c_max = math.ceil(math.log(n, 3))+1
    d_max = math.ceil(math.log(n, 2))+1

    for a in reversed(range(0,a_max)):
        aa = 7 ** a

        # n 以下の小さな合成数 7a なら、とりあえず n から、それを引く
        if aa <= n:
            remain = n - aa

            if debug:
                print(f"(7x{a}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [a, 0, 0, 0]

            # 余った数で続きをやる

            for b in reversed(range(0,b_max)):
                bb = aa * 5 ** b

                if bb <= n:
                    remain = n - bb

                    if debug:
                        print(f"\t(7x{a} x 5x{b}) remain:{remain}")

                    # 割り切れた
                    if remain == 0:
                        return [a, b, 0, 0]

                    # 余った数で続きをやる

                    for c in reversed(range(0,c_max)):
                        cc = bb * 3 ** c

                        if cc <= n:
                            remain = n - cc

                            if debug:
                                print(f"\t\t(7x{a} x 5x{b} x 3x{c}) remain:{remain}")

                            # 割り切れた
                            if remain == 0:
                                return [a, b, c, 0]

                            # 余った数で続きをやる

                            for d in reversed(range(0,d_max)):
                                dd = cc * 2 ** d

                                if dd <= n:
                                    remain = n - dd

                                    if debug:
                                        print(f"\t\t\t(7x{a} x 5x{b} x 3x{c} x 2x{d}) remain:{remain}")

                                    # 割り切れた
                                    if remain == 0:
                                        return [a, b, c, d]


if __name__ == "__main__":

    print("""\
ある正の整数 n があるとする。
とりあえず n は 1～100 ぐらいを考えておく。

n を　2, 3, 5, 7　の素数で素因数分解する。

答えがいくつかあるとき、大きな数を多く使ったものを　１つ　選ぶことにする。
""")

    print("Please input number:")
    n = int(input())

    print(f"Anser:{factorize(n)}")

「　👆　だいぶ速くなったけど、まだ遅い。一瞬で終わってほしいぜ」

「　累乗が遅くないか？　最初に a_max を指数にして累乗を求めて、　ループの終わりで　7　で割っていったらどうだぜ？」

「　速度を比較してみるか。 Python でストップウォッチの書き方は？」

📖　How to start stopwatch in python [duplicate]

import time

start = time.time()
# <code to time>
end = time.time()

print(f"Time taken to run the code was {end-start} seconds")

「　よっしゃ」

# 素因数分解の練習
#
# cd math/prime_factorization
# python step3_0.py

import time
#from step1 import factorize
from step1_1_0 import factorize
#from step1_2_0 import factorize


start = time.time()
n = 1

for i in range(0,1001):
    # 答えが出るまで繰り返す
    while True:
        answer = factorize(n, debug=False)

        if answer is not None:
            print(f"({i}) n={n} Answer={answer}")

        n += 1

        if answer is not None:
            break

end = time.time()
print(f"Time: {end-start} seconds")

「　👆　これで時間計測だ。さっきのソースは　Time: 108.6974413394928 seconds　だった」

# 素因数分解の練習
#
# cd math/prime_factorization
# python step1_2_0.py

import math

def factorize(n, debug=True):

    # int( ... ) だと、取りこぼしてしまうようだ。 math.ceil( ... ) とした
    # + 1 が必要
    a_max = math.ceil(math.log(n, 7))+1
    b_max = math.ceil(math.log(n, 5))+1
    c_max = math.ceil(math.log(n, 3))+1
    d_max = math.ceil(math.log(n, 2))+1

    aa = 7 ** a_max
    for a in reversed(range(0,a_max)):
        aa /= 7

        # n 以下の小さな合成数 7a なら、とりあえず n から、それを引く
        if aa <= n:
            remain = n - aa

            if debug:
                print(f"(7x{a}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [a, 0, 0, 0]

            # 余った数で続きをやる

            bb = 5 ** b_max
            for b in reversed(range(0,b_max)):
                bb /= 5
                aabb = aa * bb

                if aabb <= n:
                    remain = n - aabb

                    if debug:
                        print(f"\t(7x{a} x 5x{b}) remain:{remain}")

                    # 割り切れた
                    if remain == 0:
                        return [a, b, 0, 0]

                    # 余った数で続きをやる

                    cc = 3 ** c_max
                    for c in reversed(range(0,c_max)):
                        cc /= 3
                        aabbcc = aabb * cc

                        if aabbcc <= n:
                            remain = n - aabbcc

                            if debug:
                                print(f"\t\t(7x{a} x 5x{b} x 3x{c}) remain:{remain}")

                            # 割り切れた
                            if remain == 0:
                                return [a, b, c, 0]

                            # 余った数で続きをやる

                            dd = 2 ** d_max
                            for d in reversed(range(0,d_max)):
                                dd /= 2
                                aabbccdd = aabbcc * dd

                                if aabbccdd <= n:
                                    remain = n - aabbccdd

                                    if debug:
                                        print(f"\t\t\t(7x{a} x 5x{b} x 3x{c} x 2x{d}) remain:{remain}")

                                    # 割り切れた
                                    if remain == 0:
                                        return [a, b, c, d]


if __name__ == "__main__":

    print("""\
ある正の整数 n があるとする。
とりあえず n は 1～100 ぐらいを考えておく。

n を　2, 3, 5, 7　の素数で素因数分解する。

答えがいくつかあるとき、大きな数を多く使ったものを　１つ　選ぶことにする。
""")

    print("Please input number:")
    n = int(input())

    print(f"Anser:{factorize(n)}")

「　👆　このソースは　Time: 138.23809385299683 seconds、　遅くなってるなあ」

「　状態なんか持ったら　遅いか」

# 素因数分解の練習
#
# cd math/prime_factorization
# python step1_1_1_0.py

import math

def factorize(n, debug=True):

    # int( ... ) だと、取りこぼしてしまうようだ。 math.ceil( ... ) とした
    # + 1 が必要
    a_max = math.ceil(math.log(n, 7))+1
    b_max = math.ceil(math.log(n, 5))+1
    c_max = math.ceil(math.log(n, 3))+1
    d_max = math.ceil(math.log(n, 2))+1

    reversed_a_list = list(reversed(range(0,a_max)))
    reversed_b_list = list(reversed(range(0,b_max)))
    reversed_c_list = list(reversed(range(0,c_max)))
    reversed_d_list = list(reversed(range(0,d_max)))

    for a in reversed_a_list:
        aa = 7 ** a

        # n 以下の小さな合成数 7a なら、とりあえず n から、それを引く
        if aa <= n:
            remain = n - aa

            if debug:
                print(f"(7x{a}) remain:{remain}")

            # 割り切れた
            if remain == 0:
                return [a, 0, 0, 0]

            # 余った数で続きをやる

            for b in reversed_b_list:
                bb = aa * 5 ** b

                if bb <= n:
                    remain = n - bb

                    if debug:
                        print(f"\t(7x{a} x 5x{b}) remain:{remain}")

                    # 割り切れた
                    if remain == 0:
                        return [a, b, 0, 0]

                    # 余った数で続きをやる

                    for c in reversed_c_list:
                        cc = bb * 3 ** c

                        if cc <= n:
                            remain = n - cc

                            if debug:
                                print(f"\t\t(7x{a} x 5x{b} x 3x{c}) remain:{remain}")

                            # 割り切れた
                            if remain == 0:
                                return [a, b, c, 0]

                            # 余った数で続きをやる

                            for d in reversed_d_list:
                                dd = cc * 2 ** d

                                if dd <= n:
                                    remain = n - dd

                                    if debug:
                                        print(f"\t\t\t(7x{a} x 5x{b} x 3x{c} x 2x{d}) remain:{remain}")

                                    # 割り切れた
                                    if remain == 0:
                                        return [a, b, c, d]


if __name__ == "__main__":

    print("""\
ある正の整数 n があるとする。
とりあえず n は 1～100 ぐらいを考えておく。

n を　2, 3, 5, 7　の素数で素因数分解する。

答えがいくつかあるとき、大きな数を多く使ったものを　１つ　選ぶことにする。
""")

    print("Please input number:")
    n = int(input())

    print(f"Anser:{factorize(n)}")

「　👆　リストの作成を　ループの外に出したら　Time: 98.08642745018005 seconds　10秒速くなったぜ」

「　ビットコイン掘ってるみたいで　不毛な計算ねえ」

計算回そう

# 素因数分解の練習
#
# cd math/prime_factorization
# python step3_1_0.py

import time
#from step1 import factorize
#from step1_1_0 import factorize
from step1_1_1_0 import factorize # これが高速
#from step1_2_0 import factorize

print("Start")

start = time.time()
n = 1

with open('data_time.txt', mode='w') as f_t:
    with open('data.txt', mode='w') as f:

        f.write("""\
     i,          n,    7,    5,    3,    2
------, ----------, ----, ----, ----, ----
""")

        for i in range(0,1000001):
            # 答えが出るまで繰り返す
            while True:
                answer = factorize(n, debug=False)

                if answer is not None:
                    f.write(f"{i:6}, {n:10}, {answer[0]:4}, {answer[1]:4}, {answer[2]:4}, {answer[3]:4}\n")

                n += 1

                if answer is not None:
                    break

            # 100件ごとにタイムを入れる
            if i % 100 == 0:
                end = time.time()
                f_t.write(f"# (i={i}) Time: {end-start} seconds\n")
                f_t.flush()     # バッファーが溜まると書き出されるが、とりあえず毎回書きだす

        end = time.time()
        f_t.write(f"# (Finished) Time: {end-start} seconds\n")

print("Finished")

「　👆　出力は、CSV フォーマットにしよ」

「　終わったら、テキストファイルに保存したいな。
ログみたいに　垂れ流した方がいいか」

「　処理時間も出力しよ」

「　１００万件　計算するようにして、　これを寝ている間に実行しようぜ？」

ホットケーキ算☆（＾～＾）＜完全なる理解＞

2019-09-21T12:21:31+09:00

くそ記事 - Thought process

「　くそ記事を濫造しようぜ☆？」

「　お父んの頭が　おかしくなってしまった……☆」

「　物理的に　頭がおかしくなったんじゃない？」

「　サイズを気にしない人類がいるとするぜ☆
これは　１人分の　ホットケーキだぜ☆」

「　ハニーシロップを掛けないとな☆」

「　サイズは気にしないので、これも２分割とするぜ☆」

「　新しい境地ねぇ」

「　これで３分割☆　ここで、２刀断（にとうだん）という数え方も　あり　とする☆、」

「　これは　２刀断で　４分割☆」

「　誰かが研究してそうな分野ねぇ」

「　ただし　円周まで届かないカットは　禁止とする☆」

「　ここまでは　リーガル（legal; 合法）な☆」

「　おいしい　ホットケーキが　最悪の気分だぜ☆」

「　３刀断　を超えた辺りから　面白くなってくるぜ☆」

「　倍々に　ホットケーキを　損失していかない？」

「　４刀断　で　もう２０パターン☆」

「　計算方法は分かった☆　計算したくないだけで☆」

「　さっさと一般化しましょう」

「　補助線を引いてしまえば、あとは組み合わせだぜ☆」

「　あっ、被りがある☆！　このやり方では　ダメなのか……☆」

「　そこを解消しないと、次の面白い話に行けないぜ☆？」

「　じゃあ　その場しのぎで　ゼロ　で埋めておこう☆」

「　インド人もビックリだよな☆」

「　ホットケーキ算のノッテイション（Notation; 記法）が確定したところで、４刀断もやってみようぜ☆？」

「　ひとまず　こうで☆」

「　ゼロで　対応できないやつらが　いるんだが☆」

「　ゼロは　適当にバラして付ければいいんじゃないか☆？」

「　じゃあ　これで☆」

「　きふわらべが並べた順に　仮の名前を付けるぜ☆」

「　性質ごとに分けた方が　人の目には見やすいだろう☆
例えば　これで　４画　までの　ひらがな　を分類しても　面白そうだな☆」

「　こうやってみると　１か所に３本の線が重なる　ひらがな　は　無いかも知らん☆」

「　そういや　これが無いぜ☆」

「　見落としか……☆」

「　線が２本だと　２本の交点が１個☆　交点がないのが１個☆
線が３本だと　３本の交点が１個、２本の交点がが３個あるのが１個、２本の交点が２個あるのが１個、２本の交点が１個あるのが１個、交点がないのが２個☆」

「　線が４本だと
４本の交点があるのが１個、
３本の交点が１個で２本の交点が３個あるのが１個、
３本の交点が１個で２本の交点が２個あるのが１個、
３本の交点が１個で２本の交点が１個あるのが１個、
２本の交点が４個あるのが１個、
２本の交点が３個あるのが１個、
２本の交点が２個あるのが２個、
２本の交点が１個あるのが２個、
交点がないのが３個、
合計　１３個　だぜ☆」

「　ぜんぜん足りてない☆　やりなおせだぜ☆」

「　こういう形を　リストしていないことに気づいたぜ☆」

「　お父ん、不具合が見つかったぜ☆」

「　何だぜ☆？」

「　これは　分割してるとか、　刀断してるとか、言えるのかだぜ☆？」

「　してないと言えるんじゃないか☆？　例えるなら　ゼロ　のようなもの☆　答えは　ホットケーキ１枚　だよな☆」

「　端っこをカットしてるんじゃないの？　それとも　この直線は　微分の極限なの？」

「　なるほどそうか、端っこのカットだったのかだぜ☆」

「　回転は　同型なの？」

「　そう思ってやってるぜ☆」

「　ホットケーキ算の答えは　断片の枚数なの？」

「　そうだぜ☆」

「　ホットケーキ数＝刀断数の階和＋１　な気もするが……　４＋３＋２＋１＋１＝１１☆」

「　じゃあ　最大ホットケーキ数　に絞って考察を進めていこうぜ☆？」

「　台形の面積＋１　で良い気がする☆」

「　他の視点でも　見てみようぜ☆」

「　この式で合ってるのなら、このとおり作れるはずだぜ☆」

「　偶数、偶数、奇数、奇数　は　永遠にこのリズムなの？」

「　そうだぜ☆　それが　なぜかは　突き止めてある☆　説明しよう☆」

( 1* 2) =   2
( 2* 3) =   6
( 3* 4) =   12
( 4* 5) =   20

( 5* 6) =   30
( 6* 7) =   42
( 7* 8) =   56
( 8* 9) =   72

( 9*10) =   90
(10*11) =   110
(11*12) =   132
(12*13) =   156

(13*14) =   182
(14*15) =   210
(15*16) =   240
(16*17) =   272

(17*18) =   306
(18*19) =   342
(19*20) =   380
(20*21) =   420

「　n*(n+1) は　こんな感じなんだが……☆」

(  * 2) =
( 2*  ) =
(  * 4) =
( 4*  ) =

(  * 6) =
( 6*  ) =
(  * 8) =
( 8*  ) =

(  *10) =
(10*  ) =
(  *12) =
(12*  ) =

(  *14) =
(14*  ) =
(  *16) =
(16*  ) =

(  *18) =
(18*  ) =
(  *20) =
(20*  ) =

「　登場する偶数に　着目して欲しい☆」

(  * 1) =
( 1*  ) =
(  * 2) =
( 2*  ) =

(  * 3) =
( 3*  ) =
(  * 4) =
( 4*  ) =

(  * 5) =
( 5*  ) =
(  * 6) =
( 6*  ) =

(  * 7) =
( 7*  ) =
(  * 8) =
( 8*  ) =

(  * 9) =
( 9*  ) =
(  *10) =
(10*  ) =

「　あとで２で割られるので　先に割ると　こうだが、
掛ける相手は奇数なので、　奇数×奇数＝奇数、　奇数×偶数＝偶数、
そして　最後に 1 足されることを考えると　偶数、偶数、奇数、奇数☆」

「　えっ☆！？　こんな　細いところ　通すのかだぜ☆！？」

「　なんか見えてきたな☆」

「　作り方が　なんか　見えた気がするわね」

「　最初の２本は　中心から左下の方へ４５°進んだところに、微小の断片を作ることで確定なのよ」

「　これ　ホットケーキだよな……☆」

「　次の２本は　ｘ軸とｙ軸の先っぽに　微小な断片を作るように三角形を作りなさい。このとき　原点に隣接する四辺形ができているのを確認なさい」

「　ガチで　ホットケーキ　切られ始めた……☆」

「　次の２本は　ｘ軸とｙ軸の先っぽの方にできあがった焦点よりも内側を通るようになさい」

「　楽しい楽しい　ホットケーキ……☆」

「　こうしてみると！」

「　１つの平方数、２つの階和、そして＋１　の４つで構成されていることが　つまびらかに　明らかになったわね！
完全なる理解　エブリシング　ビカムズ　クリアー！」

「　おいしい　おいしい　ホットケーキ……☆」

あとがき

「　すべてを見通す眼　パーフェクト・ホットケーキ・アイズ　を手に入れた今、
最大ホットケーキ数　以外の数　を見ても　何か見えるのだろうか……☆　試してみようぜ☆？」

「　みんな頭を打ってしまったんだろうか☆？」

「　赤線と　緑色の線が　平行になると　桃色か水色のどっちでもいいんだが　象限が１つ消えてしまうんだな☆」

「　どの２軸を取るかで　だいぶ　変わるんじゃないの？」

「　雰囲気で白い曲線にしてみたが　水色とオレンジ色の　刀断の線だぜ☆」

「　２２パターンしかないのか、２２だとしたら　なぜ２２なのか　調査してみるのも　いいかもしれないな☆」

「　そうかそうか……、さらに面白い考察も可能だぜ☆」

「　色を塗り分けると　２種類の表現をもつ合同　があるように見えるが……☆」

「　階段と　１つの四角　に違いないぜ☆」

「　台形の面積＋１　だからな☆」

「　以前は　円周に隣接する断片と、円周に隣接しない断片に着目して　桃色と　黄緑の２色に塗り分けたが、
今なら　別の視点で　４色に塗り分けることができるな☆」

「　パーフェクト・ホットケーキ・アイズ　か☆」

「　人の目だと、そこに境界線を引こうとしないわよね。
階段と　１ブロック　の間に線を引きたくなるし、４×４の方眼紙から２マス欠けているのも気になるわ」

「　この形を　すっきりしている　と受け入れられるかどうかは、ホットケーキを切った者しか　考えようともしないだろうな☆」

参考：最大ホットケーキ数　2, 4, 7, 11, 16, 22, 29, 37, 46, 56, 67, 79, 92, 106, 121, 137, 154, 172 ...

＜完結＞

だらだらしようぜ☆（＾～＾）

2019-08-24T11:37:06+09:00

「　頭痛てー☆　風邪だぜ☆　だらだらしようぜ☆？」

「　寝てればいいのに……☆」

「　もう技術と関係ないんだけど」

「　サイコロの話しでもしよう☆」

「　数学の話しをするときは　パースを付けなくていいのに……☆」

「　じゃあ　そのうち　１つを　ハズレ　としよう☆」

「　１回　サイコロを振れば　外れる確率は　６分の１　だぜ☆」

「　ふつう　確率の話しをしているときは　アタリ　の話しなんで、
６分の５　と言い直そう☆」

「　ここまで　むずかしいことは　言ってないと思う☆
ゴホッ　ゴホッ　ゲェェホッ☆」

「　寝ろ☆」

次の日

「　夏休みは初日から風邪を引いているのが　頭が痛い☆
６分の５のサイコロが２つあるとしよう☆」

「　このとき、どちらも　×　でない　アタリ　の確率、
あるいは　どちらかが　×　な　ハズレ　の確率は　どれぐらいだろうか☆？」

「　×は　１個でも出てきたら　ハズレ　なの？
アタリは　×　じゃない面が１個　出てきても　アタリじゃなくなってしまうの？
ルールが非対称じゃない？」

「　アタリとは何か、ハズレとは何か、といった説明は　数学はしてくれない☆
知ってるだろ、みたいな感じで　話しは進む☆」

「　それより　お父んは　寝ろ☆」

「　話しを進めよう☆
数学の確率やるなら　サイコロを投げる前に気にすることが　いくつかある☆」

「　このサイコロは　片方が決まると　もう片方のサイコロの出目に　何か変化がでるような
決まりがあるか、ないか　だぜ☆
つまり　サイコロは、　独立事象 か、それとも独立事象ではなく 従属事象 か☆」

「　独立事象だぜ☆」

「　独立事象　であれば　並列　に書ける☆
この図は適当だが、　電気の並列つなぎ、論理の二項演算、独立事象の確率、非同期処理、並列処理、　どの分野でも　言いたいことは同じだぜ☆」

「　別々に計算して、あとで　みんな計算が終わるまで待っておいて、
最後に　合算する、ということだぜ☆」

「　１人だけ　すばやく計算が終わって　抜け駆け　したらどうなるの？」

「　そのときは　答えが２つになるだろ☆
抜け駆けしたやつの答え、　そのあとに続いてきたやつの答え☆
しかし　上図　を見てほしい☆　答えは１つ　と明示されている☆　抜け駆けは無し　というルールがあると感じろだぜ☆」

「　抜け駆けはありか、なしか　といった説明は　数学はしてくれない☆
知ってるだろ、みたいな感じで　話しは進む☆」

「　もしサイコロが　相手の顔色をうかがう　イカサマ　サイコロ　だったら　どうなるんだぜ☆？」

「　そのときは　直列つなぎ、入れ子構造、というものになる☆
前の結果を得て、次の結果が出るやつだぜ☆」

「　どっちの　サイコロ　が先に答えを出して、他のサイコロから顔色をうかがわれる側か
わからないときがあるじゃない？」

「　それは　デターミニスティック　ではなくなった　ということだぜ☆
決定的ではなく、確率的にな話しになる☆
数式は　デターミニスティック　だが、顔色をうかがうのは　数学ではなくなる☆」

「　言ってしまえば　数学　と　確率学　ぐらいに分野は　別物だぜ☆
でも　だれも気にしないから　確率や統計は　数学の授業で教えられてしまう☆」

「　確率や　統計　がなければ　世の中の事象とか　説明できないんじゃないか☆？」

「　お前らが　世の中の事象を説明している間に　本来の数学は　どこかに行っている☆
数学がお留守だぜ☆」

「　確率の説明をするために　一覧表を作ってみよう☆
Python 言語で一覧表を作るのが楽だろう☆」

main.py

for dice1 in range(1, 7):
    for dice2 in range(1, 7):
        print("{} {}".format(dice1, dice2))

Output

1 1
1 2
1 3
1 4
1 5
1 6
2 1
2 2
(Omitted below)

「　サイコロの 1 が入っていればハズレとしよう☆
サイコロに 0 はないから、目が 2 なら両方のサイコロが 1 だぜ☆」

main.py

for dice1 in range(1, 7):
    for dice2 in range(1, 7):
        one = ""
        both = ""
        summ = dice1 + dice2
        if summ == 2:
            both = "xx"

        if dice1 == 1 or dice2 == 1:
            one = "x"
        print("{} {} {: >1} {: >2}".format(dice1, dice2, one, both))

Output

「　このとき　確率論　が言う　ハズレ　というのは　×　が１個でも付いているところだぜ☆　１１個ある☆」

「　確率論がいう　アタリ　と言うのは　×　が１個も付いていないところだぜ☆　２５個ある☆」

「　５／６　×　５／６　＝　２５／３６　だな☆」

「　×　が２つ付いているところには　何か賞品はないの？」

「　確率論　は　ファンブル　なんか気にしない☆」

「　１／６　×　１／６　＝　１／３６　だな☆」

「　でも　ファンブル　って気にならない？」

「　全部外れる確率　というのは　数学用語になっていない☆
たたし　数式は立てることができる☆」

「　例えばアタリの確率のことを Rate の R という記号でも使うとしよう☆
全部外れる確率を Probability の P とする☆」

「　サイコロが１つなら　この通り☆」

「　サイコロが２つなら　この通りで……☆」

「　指数ノーテーションに書き換える☆　ペンを振る手間を減らさないと　数学　は発展しないからな☆」

「　全部外れる確率を出したいときは　１－Ｒ　というのも　何度も書かないといけない☆
当たる確率が分かれば　外れる確率も分かるから　変数は少ない方がいい☆」

「　(1-R) っていう変な式は　変数を少なくしたかったからなのね。変数を増やして何が悪いの？」

「　分かってる数を　変数にするなだぜ☆
計算機科学の一時変数みたいに　気軽に使うなだぜ☆」

じゃあ有名な式をやろうぜ☆（＾～＾）？

「　有名なところでは　ゲーム研究者の　ゼビウスの遠藤さんがブログにも書いているように、
１００人が　１００分の１が当たりのクジを１００回引くと、　６０人は何回か当たっているが、４０人は１回も当たっていない☆」

「　わらう☆」

「　わたしも　この話はしたが　数学ブログが Core Server のデータベース更新で全部消えたんで　Crieit でもう１回書く☆」

「　わらう☆」

「　１００分の１　で当たるクジは、　当たる確率Ｒ　が　１００分の１　だぜ☆　同義反復☆」

「　クジを１回引いて　当たらない確率は　１－Ｒ　☆」

「　クジを１００回引くなら　こう☆」

「　Ｒは　１００分の１　だから　こういう計算式をプログラムすれば　答えは出るわけだが……☆」

「　この形は　ネイピア数　の数式で出てくる☆」

ネイピア数

main-2.py

p = (1-1/100) ** 100
print("p = {}".format(p))

Output

p = 0.3660323412732292

「　３６人ぐらいは　ファンブル　してるわけだな☆」

「　この数は……☆」

main-2.py

print("p = {}".format(100/2.71828))

Output

p = 36.787968862663156

「　100を　ネイピア数で割ったのと　だいたい同じだぜ☆」

「　ネイピア数で割るというのが　分けわかんないんだけど」

「　それは　ネイピア数は関係なく、　割り算　が分かってないんだぜ☆
割り算　が分からないのは　国語の問題☆」

「　掛け算というのは　ズーム　なのだった☆
１が現在で、パーセンテージというのが　将来だとしよう☆」

「　100 × 0.36　は　36　だな☆」

「　割り算というのは
パーセンテージが現在で、１が　将来のようなものだぜ☆」

「　36 ÷ 0.36　は　100　だな☆」

「　じゃあ　ネイピア数で割るというのは、
現在の１００は　全体のうち　ネイピア数で掛けらて縮んだ数　になっていて」

「　ネイピア数で割ることによって
縮む前の　３６．８弱　が出てきた、ということなの？　何それ　分かんない」

「　伸びる　とか　縮む　という言葉を使うから　分からなくなる☆
フィルターを掛ける、フィルターを外す、みたいな言葉を使えだぜ☆」

「　ネイピア数というフィルターを外すことによって
１００　が　３６．８弱　になった、ということは
なんか　すごい　でかく見えるフィルターが　かかっていて　１００　に見えていた、ということなの？」

「　そう☆
まあ　2.71828 ちょい　なんだがな☆」

「　３６．８弱　が　ファンブルする人を百分率で見るフィルター　とすれば、
ネイピア数というのは　ファンブルした人の数から　クジに挑戦した人の数を見るフィルターだな☆」

「　ややこしい！」

「　数学で頻出するのは　フィルターを掛ける算、　フィルターを外す算　だからな☆
現在の数が何か、どんなフィルターがかかっているか、フィルターがかかる前の元の数が何だったか、
これは　ぱっぱ　と　できたい☆」

やってみてくれだぜ☆

「　じゃあ　１００人で　１００回、　１００分の１で当たるクジを引いてくれだぜ☆」

「　１００００回やれば　いいだけか☆
Python に厳密な乱数とかあるかだぜ☆？」

各言語での、本当に安全な乱数の作り方

main.py

import os
import sys
import random

rng = random.SystemRandom()
print(rng.randint(0, sys.maxsize))
for i in range(0, 100):
    print(rng.randrange(0, 2))

Output

2946319965679186318
0
1
0
0
0
1
1
1
1
0
1
1
0
(Omitted below)

「　まあ多分これを使って　クジを作ればいいだろう……☆」

main.py

import os
import sys
import random


def is_hit():
    rng = random.SystemRandom()
    lottery = rng.randrange(0, 100)
    return lottery == 0


person = [0] * 100

for challenge in range(0, 100):
    for i in range(0, 100):
        if is_hit():
            person[i] += 1

distribution = {}

print("""
+
| Lottery
+
""")
for i in range(0, 100):
    hits = person[i]
    print("[{}] {}".format(i, hits))

    if not hits in distribution:
        distribution[hits] = 1
    else:
        distribution[hits] += 1

print("""
+
| Distribution
+
""")
for k, v in distribution.items():
    print("[{}] {}".format(k, v))

Output

+        
| Lottery
+        

[0] 1
[1] 0
[2] 0
[3] 1
[4] 1
[5] 1
[6] 2
[7] 1
[8] 0
[9] 2
[10] 1
[11] 1
[12] 3
[13] 0
[14] 1
[15] 0
[16] 2
[17] 2
[18] 0
[19] 1
[20] 1
[21] 4
[22] 0
[23] 0
[24] 2
[25] 1
[26] 3
[27] 1
[28] 1
[29] 0
[30] 1
[31] 0
[32] 1
[33] 4
[34] 1
[35] 0
[36] 2
[37] 4
[38] 0
[39] 0
[40] 0
[41] 0
[42] 2
[43] 1
[44] 1
[45] 0
[46] 1
[47] 1
[48] 3
[49] 0
[50] 1
[51] 1
[52] 0
[53] 1
[54] 0
[55] 2
[56] 0
[57] 1
[58] 0
[59] 0
[60] 2
[61] 1
[62] 1
[63] 3
[64] 0
[65] 2
[66] 1
[67] 1
[68] 0
[69] 0
[70] 4
[71] 2
[72] 2
[73] 2
[74] 3
[75] 1
[76] 0
[77] 0
[78] 3
[79] 0
[80] 1
[81] 0
[82] 1
[83] 2
[84] 1
[85] 0
[86] 1
[87] 1
[88] 0
[89] 1
[90] 2
[91] 2
[92] 0
[93] 1
[94] 0
[95] 2
[96] 0
[97] 1
[98] 1
[99] 1

+
| Distribution
+

[1] 39
[0] 34
[2] 17
[3] 6
[4] 4

「　３４人は　１回も当たらず、　４回当たってるのが　４人いるぜ☆」

「　これを１００回やって　分布の分布を見ようぜ☆？」

「　嫌だぜ　めんどくさい……☆」

「　……☆」

main-4.py

import os
import sys
import random


def get_distribution():

    def is_hit():
        rng = random.SystemRandom()
        lottery = rng.randrange(0, 100)
        return lottery == 0

    person = [0] * 100

    for _challenge in range(0, 100):
        for i in range(0, 100):
            if is_hit():
                person[i] += 1

    distribution = {}

    for i in range(0, 100):
        hits = person[i]
        if not hits in distribution:
            distribution[hits] = 1
        else:
            distribution[hits] += 1

    return distribution


distribution_of_dist = {}

for i in range(0, 100):
    dist = get_distribution()
    for k, v in dist.items():
        if not k in distribution_of_dist:
            distribution_of_dist[k] = v
        else:
            distribution_of_dist[k] += v

    print("""
    +
    | Distribution of distribution
    +
    """)
    for k, v in distribution_of_dist.items():
        print("[{}] {}".format(k, v))

Output

(Omitted above)
    +
    | Distribution of distribution
    +

[0] 3642
[3] 666
[1] 3705
[4] 165
[2] 1791
[6] 4
[5] 27

「　累計ならこう☆　数をふやすごとに　ファンブルした数の先頭３桁が 367 ぐらいに近づくだろう☆」

＜書きかけ＞