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

Artūras Dubickas;


In this note we consider the sequence of rational numbers $b_n=\sum_{k=1}^n 2^k/k$. We show that the power of $2$ in the expansion of $b_n$ is unusually large, at least $n+1-\log_2(n+1)$, and that this bound is best possible. The sequence $b_n$, $n=1,2,3,\dots$, is related to the sequence A0031449 in the On-Line Encyclopedia of Integer Sequences.

Vol. 25 (2024), No. 1, pp. 203-208

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