Problem 241
Perfection Quotients
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)= $\frac{\sigma(n)}{n}$
Find the sum of all positive integers n ≤ 1018 for which p(n) has the form k + 1⁄2, where k is an integer.
完全度商数
对于正整数n，记σ(n)为n的所有约数之和，例如σ(6) = 1 + 2 + 3 + 6 = 12。
你可能已经知道，所谓完全数就是满足σ(n) = 2n的数。
我们定义正整数的完全度商数为p(n)= $\frac{\sigma(n)}{n}$。
在所有正整数n ≤ 1018中，有些数的p(n)可以写成k + 1⁄2，其中k是整数。求所有这样的数之和。