On Integers for Which the Sum of Divisors is the Square of the Squarefree Core

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. We show that the only solution n with at most four distinct prime factors is n = 1782. We show that there is no solution which is fourth power free. We also show that the number of solutions up to x > 1 is at most x ⅟⁴⁺ᵉ for any ε > 0 and all x > xε. Further, call n primitive if no proper unitary divisor d of n satisfies σ(d) | γ(d) ² . We show that the number of primitive solutions to the equation up to x is less than xᵉ for x > xₑ.