The Prime Sigma Function of n is the sum of the total number of positive
divisors of n.

See: http://www.oocities.org/hjsmithh/Perfect/AliquotP.html

To compute Sig(n), first factor n into its prime factors. For this document the following notation is used. An integral number n > 1 can be written as

where the Pi's are the various different prime factors, Ai the number of times Pi occurs in the prime factorization and r the number of prime factors.

The formula for Sigma is:

For example, n = 60 = 2^2 * 3^1 * 5^1

Sig(60) = (2^3 − 1)/(2 − 1) * (3^2 − 1)/(3 − 1) * (5^2 − 1)/(5 − 1)
= 7/1 * 8/2 + 24/4 = 7 * 4 * 6 = 168.

As a check, the 12 divisors are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 and they
do sum to 168.

The function Sig0(n) = Sig(n) − n is also defined so we can say that a number is a Perfect number if anf only if n = Sig0(n).

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Changes last made on Monday, 06-Aug-07 20:47:30 PDT