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单词 FormulaForSumOfDivisors
释义

formula for sum of divisors


If one knows the factorization of a number,one can compute the sum of the positive divisorsMathworldPlanetmathPlanetmath 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 seriesMathworldPlanetmath.

Theorem 1.

Suppose that n is a positive integer whose factorizationinto prime factorsMathworldPlanetmath is i=1kpimi,where the pi’s are distinct primes and themultiplicities mi are all at least 1. Thenthe sum of the divisors of n equals

i=1kpimi+1-1pi-1

and the sum of the proper divisors of n equals

i=1kpimi+1-1pi-1-i=1kpimi.
Proof.

A number will divide n if and only if primefactors are also prime factors of n andtheir multiplicity is less than to or equalto their multiplicities in n. In otherwords, a divisors n can be expressedas i=1kpiμi where 0μimi. Then the sum over all divisorsbecomes the sum over all possible choicesfor the μi’s:

dnd=0μimii=1kpiμi

This sum may be expressed as a multiplesum like so:

μ1=0m1μ2=0m2μk=0mki=1kpiμi

This sum of products may be factored intoa product of sums:

i=1k(μi=0mipiμi)

Each of these sums is a geometric series;hence we may use the formula for sum of ageometric series to conclude

dnd=i=1kpimi+1-1pi-1.

If we want only proper divisors, we shouldnot include n in the sum, so we obtainthe formula for proper divisors by subtractingn from our formula.

As an illustration, let us compute the sumof the divisors of 1800. The factorizationof our number is 233252.Therefore, the sum of its divisors equals

(24-12-1)(33-13-1)(53-15-1)=152612424=6045.

The sum of the proper divisors equals6045-1800=4245,  so we see that1800 is an abundant number.

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更新时间:2025/5/4 9:48:53