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Problem 455

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Powers With Trailing Digits

Let f(n) be the largest positive integer x less than 10⁹ such that the last 9 digits of n^x form the number x (including leading zeros), or zero if no such integer exists.

For example:

• f(4) = 411728896 (4^411728896 = …490411728896)
• f(10) = 0
• f(157) = 743757 (157⁷⁴³⁷⁵⁷ = …567000743757)
• Σf(n), 2 ≤ n ≤ 10³ = 442530011399

Find Σf(n), 2 ≤ n ≤ 10⁶.

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尾数与指数相同的幂

对于任意整数n，记f(n)是小于10⁹且满足n^x的最后9位数字恰好等于x的最大的数。如果这样的数不存在，则f(n)等于0。

例如：

• f(4) = 411728896 (4^411728896 = …490411728896)
• f(10) = 0
• f(157) = 743757 (157⁷⁴³⁷⁵⁷ = …567000743757)
• Σf(n), 2 ≤ n ≤ 10³ = 442530011399

求Σf(n), 2 ≤ n ≤ 10⁶。

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