An Algorithm to Find Sums of Consecutive Powers of Primes

04/22/2022

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by   Cathal O'Sullivan, et al.

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We present and analyze an algorithm to enumerate all integers n≤ x that can be written as the sum of consecutive kth powers of primes, for k>1. We show that the number of such integers n is asymptotically bounded by a constant times c_k x^2/(k+1)/ (log x)^2k/(k+1), where c_k is a constant depending solely on k, roughly k^2 in magnitude. This also bounds the asymptotic running time of our algorithm. We also present some computational results, using our algorithm, that imply this bound is, at worst, off by a constant factor near 0.6. Our work extends the previous work by Tongsomporn, Wananiyakul, and Steuding (2022) who examined consecutive sums of squares of primes.