Sum Of Primes Is Power Of Primes: Question

[Leroy Quet]

I just submitted these sequences:

%S A139019 1,2,5,17,103
%N A139019 a(0)=1. a(n) = smallest prime > a(n-1)
such that (sum{k=0 to n} a(k)) is a power of a
prime.
%Y A139019 A139020,A139021
%O A139019 0
%K A139019 ,more,nonn,

%S A139020 1,3,8,25,128
%N A139020 a(n) = sum{k=0 to n} A139019(k)).
%C A139020 By definition, every term is a power
of a prime.
For all n >= 0, a(2n) = 2^m, m >= 0.
%Y A139020 A139019,A139022
%O A139020 0
%K A139020 ,more,nonn,

(A139021 and A139022 are similarly defined as the
sequences above, except that A139021(0) =
A139022(0) = 2.)

Now all terms of A139020 (and of A139022) with
even index are powers of 2, as it is easily seen.

For these sequences to be infinite, a necessary
(unless forbidden m's are never obtained) but not
sufficient condition is that there is always at
least one prime p equal to:

p = 2^k - m

for some positive integer(s) k, for any fixed odd
integer m.

But is there always a k that works for all odd m?

This question might have already been answered by
application of some variation on Dirichlet's
theorem, perhaps.

Thanks,
Leroy Quet




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