计算两个数字

northwolves

假设b(n) 为最小的$n$位数$k$,满足$k^2$展开之后$0-9$的个数恰好是$k$展开之后$0-9$的个数的两倍。易知：

$b(21)=316227766050900405218$

$b(22)=3162277660190000854095$

假设a(n) 为最大的$n$位数$k$,满足$k^2$展开之后$0-9$的个数恰好是$k$展开之后$0-9$的个数的两倍。求a(21) 和a(22)

northwolves

Largest n-digit number k such that k^2 contains exactly 2 copies of each digit of k, or -1 if no such k exists.

NAME

Largest n-digit number k such that k^2 contains exactly 2 copies of each digit of k, or -1 if no such k exists.

DATA
-1, -1, -1, -1, 72576, 725760, 9883667, 99748631, 999610254, 9999634628, 99996346280, 999985001267, 9999964653201, 99999850001267, 999999821457036, 9999998760512243, 99999998504764236, 999999998245007163, 9999999987503216456, 99999999982745061023

OFFSET
1,5

COMMENTS
If a(n) <> -1, a(n) == {0,2} (mod 9).

northwolves

我的代码计算小数字还行：

1. a[n_] := (x = 10^n - 1;
   
2.   While[x > 10^(n - 1/2),
   
3.    If[DigitCount[x^2] == 2 DigitCount[x], Return[{n, x}]];
   
4.    x -= If[Mod[x, 9] == 0, 7, 2]]; {n, -1});
   
5.    Do[Print[a[n]], {n, 10}]

复制代码

{1,-1}
{2,-1}
{3,-1}
{4,-1}
{5,72576}
{6,725760}
{7,9883667}
{8,99748631}
{9,999610254}
{10,9999634628}