formula for sum of divisors
If one knows the factorization of a number,one can compute the sum of the positive divisors ofthat number without having to write downall the divisors of that number. To dothis, one can use a formula which is obtainedby summing a geometric series
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Theorem 1.
Suppose that is a positive integer whose factorizationinto prime factors is ,where the ’s are distinct primes and themultiplicities are all at least . Thenthe sum of the divisors of equals
and the sum of the proper divisors of equals
Proof.
A number will divide if and only if primefactors are also prime factors of andtheir multiplicity is less than to or equalto their multiplicities in . In otherwords, a divisors can be expressedas where . Then the sum over all divisorsbecomes the sum over all possible choicesfor the ’s:
This sum may be expressed as a multiplesum like so:
This sum of products may be factored intoa product of sums:
Each of these sums is a geometric series;hence we may use the formula for sum of ageometric series to conclude
If we want only proper divisors, we shouldnot include in the sum, so we obtainthe formula for proper divisors by subtracting from our formula.
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As an illustration, let us compute the sumof the divisors of . The factorizationof 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.