Absence of integral abundancy resonances for consecutive integers

Prove that the set V_1={v≥2:I(v)/I(v+1)∈ℕ} is empty, equivalently, prove that I(v)/I(v+1) is not a positive integer for every integer v≥2, where I(n)=σ(n)/n is the abundancy index.

Background

For a fixed shift h, the paper defines V_h as the set of integers v≥2 for which the quotient of shifted abundancy indices I(v)/I(v+h) is a positive integer. These sets govern the resonant families that can contribute regularly to the divisibility relation σ(n)∣σ(n+h).

The authors single out the consecutive case h=1 and conjecture that V_1 is empty. They note that any hypothetical v∈V_1 would satisfy v∣σ(v), and therefore would be a multiperfect number. If the conjecture holds, the paper’s dichotomy would yield a subexponential upper bound for the number of n≤x satisfying σ(n)∣σ(n+1).

References

The exceptional set introduced in Section~\ref{sec:divisibility} suggests a second, more specific conjecture. One has $\mathcal V_1=\varnothing$. Equivalently, \frac{I(v)}{I(v+1)}\notin\qquad(v\ge2).

— On the Divisibility Relation $σ(n)\midσ(n+h)$ and a Generalized Erdős--Sierpiński Conjecture  (2609.04980 - Fatehizadeh et al., 4 Sep 2026) in Conjecture 2 in Section 6.3, “Conjectures and an open problem”

\begin{openproblem}\label{prob:characterize-Vh} Characterize the shifts $h\ge1$ for which $\mathcal V_h\ne\varnothing$. \end{openproblem}

— On the Divisibility Relation $σ(n)\midσ(n+h)$ and a Generalized Erdős--Sierpiński Conjecture  (2609.04980 - Fatehizadeh et al., 4 Sep 2026) in Open Problem 1 in Section 6.3, “Conjectures and an open problem”