## The Sigma Function

### Definition

**σ(n) is the sum of the factors of n.**

**σ**_{k}(n) is the sum of the n's factors, each to the power k.
So, in particular, σ_{0}(n), sometimes called τ(n), tau(n), is the
number of factors of n.

### Working with the Sigma Function

σ(1)=1.

If p is prime then σ(p)=p+1.

If p is prime then σ(p^{n}) = 1 + p + p^{2} + ... + p^{k}
= (p^{n+1}-1)/(p-1).

Put more simply, when n is a power of a prime, p, then σ(n) = (pn-1)/(p-1)

Sigma is a multiplicative function -- that means when a and b are coprime (*i.e.*
GCD(a,b)=1), then σ(ab) = σ(a) σ(b).

Together, these rules enable you to find the sum of factors of any number, as
long as you know its prime factorization.

__Example__

17640 = 2^{3} 3^{2} 5 7^{2}

σ(17640) = (2^{4}-1)/(2-1) (3^{3}-1)/(3-1) (5^{2}-1)/(5-1)
(7^{3}-1)/(7-1)

σ(17640) = 15 * 13 * 6 * 57 = 66690

### Internet references

Mathworld:
Divisor Function

### Related Pages in this website

Totient Function -- the
number of coprimes of n.

Highly Composite Numbers
-- numbers, n, such that no number smaller than n has as many divisors as n.

The webmaster and author of this Math Help site is
Graeme McRae.