On a problem of De Koninck

Abstract

Let $\sigma (n)$ and $\gamma (n)$ denote the sum of divisors and the product of distinct prime divisors of $n$ respectively. We shall show that, if $n\ne 1,1782$ and $\sigma (n)=(\gamma (n){)}^2$, 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 $p,p^{\prime }\mid \mid n$, $q_i^2\mid \mid n(i=1,2,\dots ,k)$ and $q_1\mid \sigma (p^2),q_{i+1}\mid \sigma (q_i^2)(1\le i\le k-1),p^{\prime }\mid \sigma (q_k^2)$.

15 pages, the author's final version, address changed due to the location change of the Minoh Campus of Osaka University, to appear in Moscow J. Combin. Number Theory