Integer ratios of consecutive alternating power sums

Abstract

We give a characterization of all pairs $(k,n)$ of positive integers for which the ratio $$\frac{1^k-2^k+3^k-\cdots +(-1{)}^{n+1}{n}^k}{1^k-2^k+3^k-\cdots +(-1{)}^n(n-1{)}^k}$$ of two consecutive alternating power sums is an integer.

4 pages; to appear in the Amer. Math. Monthly