# On a problem of De Koninck

## TL;DR

This paper investigates the solutions to the equation involving the sum of divisors and the product of prime divisors of an integer, identifying specific prime divisibility conditions for all solutions except two special cases.

## Contribution

It characterizes the structure of integers satisfying (n) = ((n))^2, revealing prime divisibility patterns and excluding all but two known solutions.

## Key findings

- Identifies all solutions n (n) = (
6(n))^2 except for n=1 and n=1782.
- Describes the prime divisibility conditions that solutions must satisfy.
- Provides a structural description of solutions involving chains of primes and divisor sums.

## 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 $n\neq 1, 1782$ and $\sigma(n)=(\gamma(n))^2$, then there exist odd (not necessarily distinct) primes $p, p^\prime$ and (not necessarily odd) distinct primes $q_i (i=1, 2, \ldots, k)$ such that $p, p^\prime\mid\mid n$, $q_i^2\mid\mid n (i=1, 2, \ldots, k)$ and $q_1\mid \sigma(p^2), q_{i+1}\mid\sigma(q_i^2) (1\leq i\leq k-1), p^\prime \mid\sigma(q_k^2)$.