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)=1ᵐᵏ+2ᵐᵏ+⋯+kᵐᵏ 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.