[seqfan] Re: sigma_2(n)*sigma_3(n)/sigma(n)

[Paul Hanna]

Franklin,
      It appears that the following is an integer series:
exp( Sum_{n>=1}  sigma_2(n)*sigma_3(n)/sigma(n) * x^n/n )  = 1 + x + 8*x^2
+ 31*x^3 + 110*x^4 + 380*x^5 + 1258*x^6 + 4145*x^7 + 13062*x^8 + 40549*x^9
+ 123177*x^10 + ...

Unfortunately, the sequence having that g.f. is not in the OEIS:
[1, 1, 8, 31, 110, 380, 1258, 4145, 13062, 40549, 123177, 367524, 1078214,
3117641, 8889005, 25019907, 69592393, ...].

Just interjecting a related observation.
      Paul

On Thu, Oct 25, 2018 at 11:28 PM Frank Adams-watters via SeqFan <
seqfan at list.seqfan.eu> wrote:

> 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
>
> --
> Seqfan Mailing list - http://list.seqfan.eu/
>