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

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+ϵ, for any ϵ>0 and all x>xϵ.