Journal of Integer Sequences, Vol. 15 (2012), Article 12.5.5

Asymptotic Formulae for the n-th Perfect Power

Rafael Jakimczuk
División Matemática
Universidad Nacional de Luján
Buenos Aires


Let $ P_n$ be the $ n$-th perfect power. In this article we prove asymptotic formulae for $ P_n$. For example, we prove the following formula

$\displaystyle P_n=n^2-2n^{5/3}-2 n^{7/5} +\frac{13}{3}n^{4/3}-2n^{9/7}+2n^{6/5}-2n^{13/11}+o\left(n^{13/11}\right).$

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(Concerned with sequence A001597.)

Received October 1 2011; revised versions received January 14 2012; March 20 2012; May 29 2012. Published in Journal of Integer Sequences, May 29 2012.

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