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,...

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