# 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 |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

σ(n) |
1 | 3 | 4 | 7 | 6 | 12 | 8 | 15 | 13 | 18 | 12 | 28 | 14 | 24 | 24 | 31 |

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(*p*) is (^{n}*p*^{n+1}-1)/(*p*-1).

Example:

σ(2000) = σ(2^{4}5^{3}) = σ(2^{4})^{.}σ(5^{3}) = (2^{5}-1)/(2-1)^{.}(5^{4}-1)/(5-1) = 31^{.}156 = 4836.

Printed from the PrimePages <primes.utm.edu> © Chris Caldwell.