Divisibility of the Sums of the Power of Consecutive Integers

Abstract

We study the divisibility of the sums of the odd power of consecutive integers, $S(m,k)=1^{mk}+2^{mk}+\cdots+k^{mk}$ and $1^k+2^k+\cdots+n^k$ for odd integers $m$ and $k$, by using the Girard-Waring identity. Faulhaber's approach for the divisibilities is discussed. Some expressions of power sums in terms of Stirling numbers of the second kind are represented.