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Problem 749

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Near Power Sums

A positive integer, $n$, is a near power sum if there exists a positive integer, $k$, such that the sum of the $k$th powers of the digits in its decimal representation is equal to either $n+1$ or $n-1$. For example $35$ is a near power sum number because $3^2+5^2 = 34$.

Define $S(d)$ to be the sum of all near power sum numbers of $d$ digits or less.
Then $S(2) = 110$ and $S(6) = 2562701$.

Find $S(16)$.

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近幂和数

对于正整数$n$，若存在正整数$k$使得$n$的各位数字的$k$次幂之和等于$n+1$或$n-1$，则我们称$n$为近幂和数。例如，$35$是近幂和数，因为$3^2+5^2 = 34$。

记$S(d)$为所有至多$d$位数字的近幂和数之和。
已知$S(2) = 110$，$S(6) = 2562701$。

求$S(16)$。

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