We present and analyze an algorithm to enumerate all integers $n\le x$ 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 $$ c_k \frac{ x^{2/(k+1)} }{ (\log x)^{2k/(k+1)} }, $$ 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 give a lower bound of the same order of magnitude, and a very fast algorithm that counts such $n$. Our work extends the previous work by Tongsomporn, Wananiyakul, and Steuding (2022) who examined sums of squares of consecutive primes.

翻译：我们提出并分析一种算法，用于枚举所有满足 $n\le x$ 且可表示为连续素数 $k$ 次幂之和的整数 $n$，其中 $k>1$。我们证明，此类整数 $n$ 的个数渐近上界为常数乘以 $$ c_k \frac{ x^{2/(k+1)} }{ (\log x)^{2k/(k+1)} }, $$ 其中 $c_k$ 是仅依赖于 $k$ 的常数，其量级约为 $k^2$。该上界也限定了我们算法的渐近运行时间。我们还给出了相同量级的下界，以及一种能快速计数此类 $n$ 的算法。我们的工作扩展了 Tongsomporn、Wananiyakul 和 Steuding（2022 年）关于连续素数平方和的研究成果。