On a conjecture of de Koninck

Abstract

For a positive integer n, let σ(n) and γ(n) denote the sum of divisors and the product of distinct prime divisors of n, respectively. It is known that, if σ(n)=γ(n)2, then at most two exponents of odd primes are equal to 1 in the prime factorization of n. In this paper, we prove that, if σ(n)=γ(n)2 and only one exponent is equal to 1 in the prime factorization of n, then (1) n is divisible by 3; (2) n is divisible by the fourth powers of at least two odd primes; (3) at least two exponents of odd primes are equal to 2. We also prove that, if σ(n)=γ(n)2, then at least half of the exponents α of the primes have the property that the numbers α+1 must be either primes or prime squares.