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Perfection Quotients

Problem 241

Published on Saturday, 18th April 2009, 02:00 am; Solved by 588

For a positive integer n, let σ(n) be the sum of all divisors of n, so e.g. σ(6) = 1 + 2 + 3 + 6 = 12.

A perfect number, as you probably know, is a number with σ(n) = 2n.

Let us define the perfection quotient of a positive integer asp(n)
σ(n)
n
.

Find the sum of all positive integers n ≤ 1018 for which p(n) has the form k + 12, where k is an integer.