σ(n) is the sum of the factors of n.
σk(n) is the sum of the n's factors, each to the power k. So, in particular, σ0(n), sometimes called τ(n), tau(n), is the number of factors of n.
If p is prime then σ(p)=p+1.
If p is prime then σ(pn) = 1 + p + p2 + ... + pk = (pn+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.
17640 = 23 32 5 72
σ(17640) = (24-1)/(2-1) (33-1)/(3-1) (52-1)/(5-1) (73-1)/(7-1)
σ(17640) = 15 * 13 * 6 * 57 = 66690
Mathworld: Divisor Function
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.