Friends of 12

TL;DR

This paper investigates integers sharing the same abundancy index as 12, providing new conditions and classifications for odd and even friends of 12 using advanced number theory methods.

Contribution

It extends Ward's methods to characterize odd and even friends of 12 with specific prime factorization properties.

Findings

Odd friends of 12 are perfect squares with at least 5 prime factors including 3.

Even friends of 12 (excluding 234) have a specific form involving primes ≥29 and certain congruences.

Provides new constraints on the structure of integers sharing the abundancy index with 12.

Abstract

A friend of 12 is a positive integer different from 12 with the same abundancy index. By enlarging the supply of methods of Ward [1], it is shown that (i) if n is an odd friend of 12, then n=m^2, where m has at least 5 distinct prime factors, including 3, and (ii) if n is an even friend of 12 other than 234, then n=2*(q^e)*(m^2), in which q is a prime greater than or equal to 29, e is a positive integer, and both q and e are congruent to 1 mod 4, and m has at least 3 distinct odd prime factors, one of which is 3, and the other, none equal to q, are greater than or equal to 29.