On the Divisibility Relation $\sigma(n)\mid\sigma(n+h)$ and a Generalized Erd\H{o}s--Sierpi\'nski Conjecture

Abstract

For each fixed positive integer $h$, we study the divisibility relation $\sigma(n)\mid\sigma(n+h)$. We isolate an explicit regular family arising from integral quotients of shifted abundancy indices and show that the complementary set satisfies a subexponential saving; in particular, the number of solutions up to $x$ is $O_h(x/(\log x)^2)$. We also study the proportionality equation $\sigma(n+h)=\lambda\sigma(n)$. For every fixed nonzero integer $h$, uniformly for all real $\lambda>0$, the number of solutions up to $x$ is $O(x/\sqrt{\log\log\log x})$, with an absolute implied constant once $x$ exceeds an $h$-dependent threshold. Finally, we give an explicit family which, under Schinzel's Hypothesis $H$, produces infinitely many solutions of $\sigma(n+1)=2\sigma(n)$; the Bateman--Horn conjecture yields a precise asymptotic for the number of members of this family up to $x$. We conjecture that $\sigma(n+h)=k\sigma(n)$ has infinitely many positive integer solutions for every fixed $h,k\ge1$.