On a problem of De Koninck

Abstract

Let $σ(n)$ and $γ(n)$ denote the sum of divisors and the product of distinct prime divisors of $n$ respectively. We shall show that, if $n\neq 1, 1782$ and $σ(n)=(γ(n))^2$, then there exist odd (not necessarily distinct) primes $p, p^\prime$ and (not necessarily odd) distinct primes $q_i (i=1, 2, \ldots, k)$ such that $p, p^\prime\mid\mid n$, $q_i^2\mid\mid n (i=1, 2, \ldots, k)$ and $q_1\mid σ(p^2), q_{i+1}\midσ(q_i^2) (1\leq i\leq k-1), p^\prime \midσ(q_k^2)$.