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[组合] 数字谜三道

本帖最后由 hbghlyj 于 2019-8-5 00:12 编辑

(1)求所有的n位数,其前i位数码组成的数是i的倍数(i=1,2...n)
(2)求所有的四位数$\overline {abcd} $,$\overline {abcd}  = \frac{{{a^c} + b}}{d}$
(3)求所有2n位数x=$a_1b_1a_2b_2...a_nb_n$,x=$a_1^{b_1}a_2^{b_2}...a_n^{b_n}$
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本帖最后由 青青子衿 于 2019-8-4 23:56 编辑
(1)求所有的n位数,其前i位数码组成的数是i的倍数(i=1,2...n)
(2) ...
hbghlyj 发表于 2019-8-4 21:11

\begin{align*}
(k_{\overset{\,}i}\in\mathbf{N}_+)\\
\\
\overline{a_{\overset{\,}1}a_{\overset{\,}2}}=2k_2\\
\overline{a_{\overset{\,}1}a_{\overset{\,}2}a_{\overset{\,}3}}=3k_3\\
\overline{a_{\overset{\,}1}a_{\overset{\,}2}a_{\overset{\,}3}a_{\overset{\,}4}}=4k_4\\
\vdots\qquad\\
\end{align*}
  1. f[2] = Select[Range[10, 99], EvenQ];
  2. f[x_] := f[x] =
  3.   Select[Flatten[10 # + Range[0, 9] & /@ f[x - 1]], Mod[#, x] == 0 &]
  4. f[3]
  5. Do[Print["满足条件", ToString[x], "位数的数目=", ToString[Length@f[x]]], {x, 2,
  6.     26}] // AbsoluteTiming
复制代码
...
满足条件2位数的数目=45
满足条件3位数的数目=150
满足条件4位数的数目=375
满足条件5位数的数目=750
满足条件6位数的数目=1200
满足条件7位数的数目=1713
满足条件8位数的数目=2227
满足条件9位数的数目=2492
满足条件10位数的数目=2492
...
【Oeis】A143671   Number of terms in A144688 of length n.
http://oeis.org/A143671
0010, 0045, 0150, 0375, 0750,
1200, 1713, 2227, 2492, 2492,
2225, 2041, 1575, 1132, 0770,
0571, 0335, 0180, 0090, 0044,
18, 12, 60, 03, 01,
00, 00, 00, 00, 00,
00, 00, 00, 00, 00,
00, 00, 00, 00, 00, ...

Total: 20457

\begin{align*}
36&=\phantom{0}2\cdot18\\
360&=\phantom{0}3\cdot120\\
3608&=\phantom{0}4\cdot902\\
36085&=\phantom{0}5\cdot7217\\
360852&=\phantom{0}6\cdot60142\\
3608528&=\phantom{0}7\cdot515504\\
36085288&=\phantom{0}8\cdot4510661\\
360852885&=\phantom{0}9\cdot40094765\\
3608528850&=10\cdot360852885\\
36085288503&=11\cdot3280480773\\
360852885036&=12\cdot30071073753\\
3608528850368&=13\cdot277579142336\\
36085288503684&=14\cdot2577520607406\\
360852885036840&=15\cdot24056859002456\\
3608528850368400&=16\cdot225533053148025\\
36085288503684007&=17\cdot2122664029628471\\
360852885036840078&=18\cdot20047382502046671\\
3608528850368400786&=19\cdot189922571072021094\\
36085288503684007860&=20\cdot1804264425184200393\\
360852885036840078603&=21\cdot17183470716040003743\\
3608528850368400786036&=22\cdot164024038653109126638\\
36085288503684007860367&=23\cdot1568925587116695993929\\
360852885036840078603672&=24\cdot15035536876535003275153\\
3608528850368400786036725&=25\cdot144341154014736031441469
\end{align*}

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