# Prime Sigma Function

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

n = P1^A1 * P2^A2 * . . . * Pr^Ar

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:

Sig(N) = ((P1^(A1+1) − 1) / (P1−1)) * ... * ((Pr^(Ar+1) − 1) / (Pr−1)).

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).

See: Sigma Function -- From The Prime Glossary
And: Divisor -- From MathWorld

This page accessed times since March 26, 2005.
Page created by: hjsmithh@sbcglobal.net
Changes last made on Monday, 06-Aug-07 20:47:30 PDT