formula for sum of divisors
If one knows the factorization of a number,
one can compute the sum of the positive divisors of
that number without having to write down
all the divisors of that number. To do
this, one can use a formula which is obtained
by summing a geometric series.
A number will divide if and only if prime factors are also prime factors of and their multiplicity is less than to or equal to their multiplicities in . In other words, a divisors can be expressed as where . Then the sum over all divisors becomes the sum over all possible choices for the ’s:
This sum may be expressed as a multiple sum like so:
This sum of products may be factored into a product of sums:
Each of these sums is a geometric series; hence we may use the formula for sum of a geometric series to conclude
If we want only proper divisors, we should
not include in the sum, so we obtain
the formula for proper divisors by subtracting
from our formula.
As an illustration, let us compute the sum of the divisors of . The factorization of our number is . Therefore, the sum of its divisors equals
The sum of the proper divisors equals , so we see that is an abundant number.
|Title||formula for sum of divisors|
|Date of creation||2013-03-22 16:47:35|
|Last modified on||2013-03-22 16:47:35|
|Last modified by||rspuzio (6075)|