zeta(r)^zeta(r)

[Leroy Quet]

>[my original post and reply to sci.math] 
>
>qqquet at mindspring.com (Leroy Quet) wrote in message 
>news:<b4be2fdf.0403081540.7d8e0625 at posting.google.com>...
>> Let a(1) = 1;
>> 
>> Let, for m >= 2,
>> 
>> a(m) = 
>> 
>> (1/ln(m)) sum{p=primes} sum{k|m,k>=2} a(m/k) ln(k) H(c(p,k)),
>> 
>> where H(n) = 1+1/2+1/3+...+1/n, the n_th harmonic number,
>> 
>> and c(p,k) is a nonnegative integer where
>> p^c(p,k) is the highest power of the prime p which divides k.
>> 
>> (And, oh yeah, H(0) =0.)
>> 
>> 
>> So, we have then:
>> 
>> sum{m=1 to oo} a(m)/m^r  =
>> 
>> zeta(r)^zeta(r),
>> 
>> unless I made a mistake.
>> (My math was not rigorous.)
>> 
>> But what I am wondering is,
>> what is a closed-form (non-recursive definition) for
>> a(m) ??
>> 
>
>I might as well give the first few terms of this sequence:
>
>a(m): 1, 1, 1, 2, 1, 3, 1, 7/2, 2,...
>
>Is suspect that each term is a rational which depends only on the 
>exponents in the prime-factorization of that term's index, but I am not 
>absolutely certain.
>
>If, for example, b(k) = a(p^k), p = prime, then:
>
>sum{k=0 to oo} x^k b(k)  =
>
>(1/(1-x))^(1/(1-x)),
>
>I believe.
>
>And b(k)(k-1)! forms the sequence:
>1, 2, 7, 35,...

I gave the b-sequence from k = 1, of course.

And b(k)*k! is:

http://www.research.att.com/cgi-bin/access.cgi/as/njas/sequences/eisA.cgi?A
num=A087761 


>
>(Is this sequence, or anything related to the a-sequence, in the EIS??)
> 

thanks,
Leroy Quet