A conjecture of De Koninck regarding particular square values of the sum of divisors function

Abstract

We study integers n > 1 satisfying the relation σ(n) = γ(n)², where σ(n) and γ(n) are the sum of divisors and the product of distinct primes dividing n, respectively. If the prime dividing a solution n is congruent to 3 modulo 8 then it must be greater than 41, and every solution is divisible by at least the fourth power of an odd prime. Moreover at least 2/5 of the exponents a of the primes dividing any solution have the property that a + 1 is a prime power. Lastly we prove that the number of solutions up to x > 1 is at most x¹/⁶⁺є, for any є > 0 and all x > xє.