Algorithms and Bounds on the Sums of Powers of Consecutive Primes

We present and analyze an algorithm to enumerate all integers  n≤x  that can be written as the sum of consecutive  k th 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))/ (logx)^( 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.