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

[Paul Hanna]

FYI,
      The sequence with g.f.  exp( Sum_{n>=1}
sigma_2(n)*sigma_3(n)/sigma(n) * x^n / n )
has been submitted as   https://oeis.org/A320416   with the comment:

"This sequence is conjectured to consist entirely of integers.
Related to A320917(n) = sigma_2(n)*sigma_3(n)/sigma(n)."

Best Regards to All,
       Paul

On Fri, Oct 26, 2018 at 9:27 AM Paul Hanna <pauldhanna.math at gmail.com>
wrote:

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