On a sequence of rational numbers with unusual divisibility by a power of 2

Abstract

In this note we consider the sequence of rational numbers bn = ∑k=1n2k∕k. We show that the power of 2 in the expansion of bn is unusually large, at least n + 1 − log2(n + 1), and that this bound is best possible. The sequence bn, n = 1,2,3,…, is related to the sequence A0031449 in the On-Line Encyclopedia of Integer Sequences.