[seqfan] sigma_2(n)*sigma_3(n)/sigma(n)
Frank Adams-watters
franktaw at netscape.net
Fri Oct 26 04:37:25 CEST 2018
I am submitting a new sequence, https://oeis.org/draft/A320917, with this definition. The problem is to prove that it is always an integer.
This reduces to showing that a(p^k) is a polynomial in p for fixed positive integer k, where p is a prime. This is trivially a rational function, so we need to show that that rational function, in lowest terms, has denominator = 1.
I have verified this up to k = 1024, but I don't see how to continue to an actual proof.
Failing that, it would be interesting to see what happens at k = 2^32, so k+1 is the the first non-prime Fermat number. Checking this is far beyond any brute-force computation I can make, nor do I know any way to compute it efficiently. I'm not certain that there is any tie to the Fermat Numbers; it's just an intuitive leap.
Franklin T. Adams-Watters
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