On a problem of De Koninck

摘要

Let σ(n) denote the sum of divisors of n and γ(n) denote the product of distinct prime divisors of n. We shall show that, if n ≠ 1; 1782 and σ(n) = (γ(n))~2, then there exist odd (not necessarily distinct) primes p, p' and (not necessarily odd) distinct primes q_i (i = 1, 2, ..., k) such that p, p' || n, q_i~2||n (i = 1, 2, ... , k), with k ≤ 3, and q_1 |σ(p~2), q_(i+1) | σ(q_i~2) (1≤i ≤k - 1), p' | σ (q_k~2).