On a problem of De Koninck

Abstract

Let $\sigma \left(n\right)$ denote the sum of divisors of $n$ and $\gamma \left(n\right)$ denote the product of distinct prime divisors of $n$. We shall show that, if $n\ne 1,1782$ and $\sigma \left(n\right)={\left(\gamma \left(n\right)\right)}^2$, then there exist odd (not necessarily distinct) primes $p,p^{\prime }$ and (not necessarily odd) distinct primes $q_i$ ($i=1,2,\dots ,k$) such that $p,p^{\prime }\parallel n$, $q_i^2\parallel n$ ($i=1,2,\dots ,k$), with $k\le 3$, and $q_1|\sigma \left(p^2\right)$, $q_{i+1}|\sigma \left(q_i^2\right)$ ($1\le i\le k-1$), $p^{\prime }|\sigma \left(q_k^2\right)$.