Does this sequence finite and complete?

[Max Alekseyev]

Personally, I think this sequence is neither finite, nor complete. But
the next term of this sequence may be very large. My arguments follow.

Let f(d,n) = Sum_{k=1..n} 1/k^d.

For a prime p and d be from {2,3,4,5,6}, we have
f(d,p-1) = Sum_{k=1..p-1} 1/k^d.

Let q < p be a prime number and let u=[log(p)/log(q)] so that q^u < p
< q^(u+1). Define res(p,q)=[p/q^u]. It is clear that 1 <= res(p,q) <
q.

Consider the terms of the sum representing f(d,p-1), with the smallest
valuation w.r.t. q.
The sum of such terms is
Sum_{k=1..res(p,q)} 1/(k*q^u)^d = 1/q^(u*d) * f(d,res(p,q)).

Lemma 1. If valuation(f(d,res(p,q)),q) = 0 then valuation(f(d,p-1),q) = -u*d.

For a prime p, we call prime q<p problematic if for some d from
{2,3,4,5,6}, valuation(f(d,res(p,q)),q)>0. We call a prime p strong if
there are no problematic primes for it.

Lemma 2. Strong primes belong to the sequence A127064.

I believe that it is quite unlikely for a non-strong prime to be an
element of A127064. Actually, all currently listed terms of A127064
are strong. So, it makes sense to focus on finding strong primes.

Using extensive sieving, I have got that the next strong prime is at
least 24813266100361.
Moreover, for p=24813266100361 I have proved that
* p does not have problematic primes below 4982000.
* if q is a problematic prime for p, then res(p,q)>=37 (in particular
implying that if a problematic prime q exists then q<=[p/37]).

Regards,
Max

On 1/4/07, Artur <grafix at csl.pl> wrote:
> I was contribited in ONEIS following sequence of primes:
> %S A127064 2, 3, 5, 17
> %N A127064 Primes p such that denominator of Sum_{k=1..p-1} 1/k^2} is
> square and denominator Sum_{k=1..p-1} 1/k^3} is cube and denominator
> Sum_{k=1..p-1} 1/k^4} is fourth power and denominator Sum_{k=1..p-1}
> 1/k^5} is fifth power and denominator Sum_{k=1..p-1} 1/k^6} is sixth power
> %C A127064 Does this sequence finite and complete?
> %Y A127064 Cf. A061002, A034602, A127029, A127042, A127043, A127044,
> A127046, A127047, A127048, A127049, A127051, A127061, A127062, A127063
> %O A127064 1
> %K A127064 ,nonn,
> %A A127064 Artur Jasinski (grafix at csl.pl), Jan 04 2007
>
> I'm asking: Does this sequence finite and complete?
>
> Who can prooved these hypothesis or give countersample ?
>
> ARTUR JASINSKI
>
>
>
>