determination of abundant numbers with specified prime factors

The formulaMathworldPlanetmathPlanetmath for sums of factors may be used to find all abundant numbers with a specified set of prime factorsMathworldPlanetmath or that no such numbers exist. To accomplish this, we first do a little algebraic manipulation to our formula.

Theorem 1.

A number n whose factorization into prime numbersMathworldPlanetmath is i=1kpim1 is abundant if and only if


By definition n is abundant, if the sum of the proper divisors of n is greater than n. Using our formula, this is equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath to the condition


Dividing the k-th term in the productPlanetmathPlanetmath on the left-hand side by the k-th term on the right-hand side,


so the condition becomes


Note that each of the terms in the product is bigger than 1. Furthemore, the k-th term is bounded by


This means that it is only possible to have an abundant number whose prime factors are {pi1ik} if


As it turns out, the convers also holds, so we have a nice criterion for determining when a set of prime numbers happens to be the set of prime divisors of an abundant number.

Theorem 2.

A finite setMathworldPlanetmath S of prime numbers is the set of prime divisors of an abundant number if and only if


As described above, if S is a set of prime factors of an abundant number, then we may bound each term in the inequalityMathworldPlanetmath of the previous theorem to obtain the inequality in the current theorem. Assume, then that S is a finite set of prime numbers which satisfies said inequality. Then, by continuity, there must exist a real number ϵ>0 such that


whenver 0<x<ϵ. Since limkn-k=0 when n>1, we can, for every pS, find an m(p) such that




so, by the previous theorem, pSpm(p) is an abundant number. ∎

Title determination of abundant numbers with specified prime factors
Canonical name DeterminationOfAbundantNumbersWithSpecifiedPrimeFactors
Date of creation 2013-03-22 16:47:41
Last modified on 2013-03-22 16:47:41
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 12
Author rspuzio (6075)
Entry type Theorem
Classification msc 11A05