[seqfan] Re: Unlikely Integer Sequences

Paul D Hanna pauldhanna at juno.com
Thu Dec 30 23:20:51 CET 2010

Hi Neil (and SeqFans), 
     Soon after my prior email to seqfans, Juan Arias de Reyna sent me a concise proof that the sequences were not integral. The basis of the proof is covered in one of his papers and involves Kummer congruences:  
J. Arias de Reyna, Dynamical zeta functions and Kummer congruences, Acta Arithmetica 119, (2005) 39--52. 
Seeing as they are nonintegral, I have quickly lost interest in these sequences. 
---------- Original Message ----------
From: "N. J. A. Sloane" <njas at research.att.com>
To: seqfan at list.seqfan.eu
Cc: njas at research.att.com
Subject: [seqfan] Re: Unlikely Integer Sequences
Date: Thu, 30 Dec 2010 16:14:18 -0500

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.

Best regards

>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. 
>      Paul 
>SEQ. A1: 
>G.f.: exp( Sum_{n>=1} sigma(n,2^n)*x^n/n ). 
>a(n) = (1/n)*Sum_{k=1..n} sigma(k,2^k)*a(n-k) for n>0, with a(0) = 1.
>    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 ). 
>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)))}


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