Generalized Erdős–Sierpiński infinitude conjecture

Establish that, for every pair of positive integers h and k, the equation σ(n+h)=kσ(n) has infinitely many positive integer solutions n.

Background

The paper studies shifted divisibility and proportionality relations involving the sum-of-divisors function σ. Every solution of σ(n+h)=kσ(n) is a solution of the divisibility relation σ(n)∣σ(n+h), and the paper proves quantitative upper bounds showing that, for fixed h, the number of such solutions up to x is O_h(x/(log x)^2).

Despite this sparsity, the authors conjecture that every fixed positive shift h and positive integer multiplier k admit infinitely many solutions. The conjecture extends the classical Erdős–Sierpiński problem, corresponding to (h,k)=(1,1), and is supported conditionally for at least one nontrivial case: under Schinzel’s Hypothesis H, the paper constructs infinitely many solutions to σ(n+1)=2σ(n).