Sums of perfect powers. (English) Zbl 1285.11051

Jakimczuk, Rafael

Int. J. Contemp. Math. Sci. 8, No. 1-4, 61-67 (2013).

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

Keywords:

perfect powers; sums of perfect powers