[seqfan] Unlikely Integer Sequences

Paul D Hanna pauldhanna at juno.com
Thu Dec 30 08:18:30 CET 2010

```SeqFans,
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)))}
-------------------------------------------------------------------
[END]

```