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^1/4+ε for any ε > 0 and all x > x_ε. Further, call n primitive if no proper unitary divisor d of n satisfies σ(d) | γ(n)². We show that the number of primitive solutions to the equation up to x is less than x^ε for x > x_ε.

Received February 23 2012; revised versions received July 26 2012; August 14 2012; September 3 2012. Published in Journal of Integer Sequences, September 8 2012.