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Perfection Quotients
Problem 241
Published on 18 April 2009 at 02:00 am [Server Time]
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.
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