[seqfan] Re: Unlikely Integer Sequences

N. J. A. Sloane njas at research.att.com
Thu Dec 30 22:14:18 CET 2010

```Paul, We have other examples of sequences that go nonintegral
after a while (there's an entry for them in the Index to the OEIS).
Could you go ahead and submit the terms you have (as new sequences)?
They are certainly interesting.
Thanks

Best regards
Neil

>From: "Paul D Hanna" <pauldhanna at juno.com>
>Date: Thu, 30 Dec 2010 07:18:30 GMT
>To: seqfan at list.seqfan.eu
>      Below I descibe 2 related sequences that are unlikely to consist entirely of integers.
>I have no basis for thinking that they should, but at least the initial 25 terms of each sequence are in fact integers.
>
>Could someone attempt to find the first non-integral term in these sequences?
>Or find more integer terms than 25 (warning: very large numbers are generated!).
>I attempted to do so with the programs I provide, but ran out of memory.
>THANKS,
>      Paul
>-------------------------------------------------------------------
>SEQ. A1:
>G.f.: exp( Sum_{n>=1} sigma(n,2^n)*x^n/n ).
>
>1,1,9,2196,1073760436,4656612873078466356057,
>10556714443828879618265995132510420512858395558123,...
>
>Recurrence:
>a(n) = (1/n)*Sum_{k=1..n} sigma(k,2^k)*a(n-k) for n>0, with a(0) = 1.
>
>COMMENT:
>    sigma(n,k) denotes the sum of the k-th powers of the divisors of n.
>    Compare g.f. with the g.f. of the partition numbers:
>    exp( Sum_{n>=1} sigma(n)*x^n/n ).
>
>(PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(m,2^m)*x^m/m)+x*O(x^n)),n)}
>(PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k,2^k)*a(n-k)))}
>-------------------------------------------------------------------
>SEQ. A2:
>G.f.: exp( Sum_{n>=1} sigma(n)^(2^n)*x^n/n ).
>
>1,1,41,21886,8308232665086,1591732221997588410417273,
>194737009604544410307191356478908609904736951685592460489553926559808,...
>
>Recurrence:
>a(n) = (1/n)*Sum_{k=1..n} sigma(k)^(2^k)*a(n-k) for n>0, with a(0) = 1.
>
>(PARI) {a(n)=polcoeff(exp(sum(m=1,n,sigma(m)^(2^m)*x^m/m)+x*O(x^n)),n)}
>(PARI) {a(n)=if(n==0,1,(1/n)*sum(k=1,n,sigma(k)^(2^k)*a(n-k)))}
>-------------------------------------------------------------------

```