Let c(n) be a function that counts how many positive divisors the positive integer n has, and let

S(n) := summation over d/n { c(d) }

be the sum of the counts c(d) taken over all divisors d of n.

Examples:

Ex1: 15 is divisible by 1, 3, 5, 15, so c(15) = 4. Since c(1) = 1, c(3 )= c(5) = 2, and c(15) = 4, we find that S(15) = 1+2+2+4= 9.

Ex2: 18 is divisible by 1, 2, 3, 6, 9, and 18, so c(18) = 6. Since c(1) = 1, c(2) = c(3) = 2, c(6) = 4 and c(9) = 3, we find that S(18) = 1+2+2+4+3+6 = 18.

Find all roots n of the equation S(n) = n.

Source: Putnam 2002.