## Perfection Quotients

### Problem 241

Published on Saturday, 18th April 2009, 02:00 am; Solved by 545

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 as | p(n) | = | σ(n) n | .

Find the sum of all positive integers n 1018 for which p(n) has the form k + 1⁄2, where k is an integer.