On positive integers n with $\sigma _l (2n+1)<\sigma _l (2n)$

Abstract

For any integer l and any positive integer n, let $\sigma _{l}(n)=\sum _{d\mid n}d^{l}$. In 1936, Erdős proved that the set of positive integers n with $\sigma _1 (n+1)\ge \sigma _1 (n)$ has natural density $\frac{1}{2}$. Recently, M. Kobayashi and T. Trudgian showed that the set of positive integers n with $\sigma _1 (2n+1)\ge \sigma _1 (2n)$ has natural density between 0.053 and 0.055. In this paper, for $|l|\ge 2$ we prove that $\sigma _l (2n+1)<\sigma _l (2n)$ and $\sigma _l (2n-1)<\sigma _l (2n)$ for all sufficiently large integers n. We also correct a theorem of Erdős. Two conjectures and two problems are posed for further research.