Divisibility of the sums of the power of consecutive integers

We study the divisibility of the sums of the odd power of consecutive integers, S (m, k) =1^mk+2^mk++k^mk and 1ᵏ+2ᵏ++nᵏ 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.