On the Divisibility Relation σ(n)|σ(n+h) and a Generalized Erdős--Sierpiński Conjecture
Abstract
For each fixed positive integer h, we study the divisibility relation σ(n)|σ(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 Oh(x/(log x)2). We also study the proportionality equation σ(n+h)=λσ(n). For every fixed nonzero integer h, uniformly for all real λ>0, the number of solutions up to x is O(x/√(logloglog 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 σ(n+1)=2σ(n); the Bateman--Horn conjecture yields a precise asymptotic for the number of members of this family up to x. We conjecture that σ(n+h)=kσ(n) has infinitely many positive integer solutions for every fixed h,k≥1.
Related
BibTeX
@misc{fatehizadeh2026divisibility,
title = {{On the Divisibility Relation $σ(n)\midσ(n+h)$ and a Generalized Erdős--Sierpiński Conjecture}},
author = {Amirali Fatehizadeh and Florian Luca},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04980},
eprint = {2609.04980},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04980},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the Divisibility Relation σ(n)|σ(n+h) and a Generalized Erdős--Sierpiński Conjecture
AU - Amirali Fatehizadeh
AU - Florian Luca
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04980
UR - https://arxiv.org/abs/2609.04980
AB - For each fixed positive integer h, we study the divisibility relation σ(n)|σ(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 σ(n+h)=λσ(n). For every fixed nonzero integer h, uniformly for all real λ>0, the number of solutions up to x is O(x/√(logloglog 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 σ(n+1)=2σ(n); the Bateman--Horn conjecture yields a precise asymptotic for the number of members of this family up to x. We conjecture that σ(n+h)=kσ(n) has infinitely many positive integer solutions for every fixed h,k≥1.
ER -