sigma function

The sigma function of a positive integer n is the sum of the positive divisors of n. This is usually σ(n) using the greek letter sigma.

Here are the first few values of this function:

integer n 123456 78 910111213 14 1516
σ(n) 1347 612 815 131812281424 2431

Clearly, for primes p, σ(p)=p+1. The function σ(x) is a multiplicative function, so its value can be determined from its value at the prime powers:

Theorem
If p is prime and n is any positive integer, then σ(pn) is (pn+1-1)/(p-1).

Example:

σ(2000) = σ(2453) = σ(24).σ(53) = (25-1)/(2-1) . (54-1)/(5-1) = 31 . 156 = 4836.

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