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Sums of perfect powers. (English) Zbl 1285.11051
Summary: Let \(P_n\) be the \(n\)th perfect power. In this article we obtain asymptotic formulae for the sum \(\sum_{i=1}^n P_i\). We also prove the following formulae \[ \sum_{i=1}^n \frac 1{\sqrt{P_i}}=\log n+C+o(1),\quad \sum_{P_n\leq x} \frac 1{\sqrt{P_i}}=\frac 12 \log x+C+o(1), \] where \(C\) is a constant.
MSC:
11B83 Special sequences and polynomials
11N25 Distribution of integers with specified multiplicative constraints
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