# 10.22 Sigma

`Sigma( n )`

`Sigma` returns the sum of the positive divisors (see DivisorsInt) of the integer n.

`Sigma` is a multiplicative arithmetic function, i.e., if n and m are relatively prime we have sigma(n m) = sigma(n) sigma(m). Together with the formula sigma(p^e) = (p^{e+1}-1) / (p-1) this allows you to compute sigma(n).

Integers n for which sigma(n)=2 n are called perfect. Even perfect integers are exactly of the form 2^{n-1}(2^n-1) where 2^n-1 is prime. Primes of the form 2^n-1 are called Mersenne primes, the known ones are obtained for n = 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, and 859433. It is not known whether odd perfect integers exist, however BC89 show that any such integer must have at least 300 decimal digits.

`Sigma` usually spends most of its time factoring n (see FactorsInt).

```    gap> Sigma( 0 );
Error, Sigma: <n> must not be 0
gap> Sigma( 1 );
1
gap> Sigma( 1009 );
1010        # thus 1009 is a prime
gap> Sigma( 8128 ) = 2*8128;
true        # thus 8128 is a perfect number ```

GAP 3.4.4
April 1997