Wikipedia:Reference desk/Archives/Mathematics/2024 May 10

= May 10 =

About abundance and abundancy
Let s(n) = (n) = sigma(n)-n = sum of divisors of n that are less than n

2402:7500:942:8E8F:A4D8:9B73:8E52:1E7B (talk) 07:41, 10 May 2024 (UTC)
 * 1) Give integer k, should there be infinitely many positive integers n such that s(n)-n = k?
 * 2) Give positive rational number k, should there be infinitely many positive integers n such that s(n)/n = k?


 * The answer to 1. is no. If a number has is composite, then it is completely determined by its set of proper divisors (in particular, it is the product of the smallest prime factor and the largest proper divisor.) By definition $$s(n) - n = m$$ if and only if there is a partition of $$m$$ into unique numbers such that the elements of the partition are precisely the proper divisors of $$n$$. There are a finite amount of possible partitions of $$m$$, and thus a finite number of partitions which produce the proper divisors of some number $$n$$, and as long as the partitions in question are not just the set $$\{1\}$$ (i.e. the partition produced by primes), all such partitions/sets of proper divisors completely determine some unique $$n$$. Thus for $$k \neq 1$$ there are a finite number of $$n$$ satisfying $$s(n) - n = m$$. GalacticShoe (talk) 17:28, 10 May 2024 (UTC)
 * The smallest values of $$n$$ such that $$s(n) - n = m$$ are given in A070015, while the largest values of $$n$$ such that $$s(n) - n = m$$ are given in  A135244. GalacticShoe (talk) 17:32, 10 May 2024 (UTC)
 * Well, I meant s(n)-n = sigma(n)-2*n, not sigma(n) - n (which is s(n) itself), s(n) is, while sigma(n) is , they are different functions. 2402:7500:900:DEEB:B513:C07E:8EF3:8275 (talk) 04:09, 11 May 2024 (UTC)
 * See A033880. GalacticShoe (talk) 18:45, 11 May 2024 (UTC)
 * Well, so should there be infinitely many such positive integers n? 49.217.136.82 (talk) 07:42, 14 May 2024 (UTC)
 * Unfortunately I have no idea, you're gonna have to check the sources in that OEIS listing. GalacticShoe (talk) 08:38, 14 May 2024 (UTC)