On a problem of Pongsriiam on the sum of divisors

Abstract

For any positive integer $n$, let $\sigma (n)$ be the sum of all positive divisors of $n.$ In this paper, it is proved that for every integer $1\le k\le 29,(k,30)=1,$ we have $$\sum _{n\le K}\sigma (30n)>\sum _{n\le K}\sigma (30n+k)$$ for all $K\in \mathbb{N},$ which gives a positive answer to a problem posed by Pongsriiam recently.