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