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May 10

About abundance and abundancy

Let s(n) = (sequence A001065 in the OEIS)(n) = sigma(n)-n = sum of divisors of n that are less than n

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?

2402:7500:942:8E8F:A4D8:9B73:8E52:1E7B (talk) 07:41, 10 May 2024 (UTC)

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 OEIS: A070015, while the largest values of $n$ such that $s(n)-n=m$ are given in OEIS: 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 (sequence A001065 in the OEIS), while sigma(n) is (sequence A000203 in the OEIS), they are different functions. 2402:7500:900:DEEB:B513:C07E:8EF3:8275 (talk) 04:09, 11 May 2024 (UTC)

See OEIS: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)