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Problem 63

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Powerful digit counts

The $5$-digit number, $16807=7^5$, is also a fifth power. Similarly, the $9$-digit number,
$134217728=8^9$, is a ninth power.

How many $n$-digit positive integers exist which are also an $n$th power?

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幂次与位数

五位数$16807=7^5$同时也是一个五次幂。类似地，九位数$134217728=8^9$同时也是九次幂。

有多少个$n$位正整数同时也是$n$次幂？

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