On the Problem σ_od(n) = σ_od(n + 1)

Abstract

Let $\sigma_{od}(n) =\sum_{d|n,2\nmid d} d$. In this paper, we study the solutions of $\sigma_{od}(n)=\sigma_{od}(n+1)$, their relations to Pell numbers, and some interesting conjectures. Finally, we obtain the equation

$\sigma_{od}(n)=\sigma_{od} (n+1)=\sigma_{od} (n+2) \equiv 1\pmod2$ has no solution.