## Perfection Quotients

### Problem 241

Published on Saturday, 18th April 2009, 02:00 am; Solved by 614; Difficulty rating: 80%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`) = 2`n`.

Let us define the perfection quotient of a positive integer as | p(n) | = | σ( n)n |
. |

Find the sum of all positive integers `n` ≤ 10^{18} for which `p`(`n`) has the form `k` + ^{1}⁄_{2}, where `k` is an integer.