Provable Riesel numbers (A076337).

[David Wilson]

Dani Berend essentially asks:

> Are there odd numbers n such that n+2^k is known to be
> composite for every k >= 1?

Conjecturally, this is true for the Sierpinski numbers (A076336).
It is definitely true for those n currently listed in A076336.

NJAS:

I recommend the following changes to A076336:

1.  Replace the existing title with

%N Provable Sierpinski numbers: n such that for all k >= 1 the numbers n*2^k 
+ 1 are composite.

which acknowledges that there may exist Sierpinski numbers that
are unprovable by the standard method of exhibiting a periodic
sequence of divisors.

2.  Condense the following discussion

%C A076336 There are 4 related sequences that arise in this context:
%C A076336 S1: Numbers n such that n*2^k + 1 is composite for all k (this 
sequence)
%C A076336 S2: Odd numbers n such that 2^k + n is composite for all k 
(apparently it is conjectured that S1 and S2 are \
  the same sequence)
%C A076336 S3: Numbers n such that n*2^k + 1 is prime for all k (empty)
%C A076336 S4: Numbers n such that 2^k + n is prime for all k (empty)
%C A076336 The following argument, kindly provided by Michael Reid, shows 
that S3 and S4 are empty:
%C A076336 If p is a prime divisor of n + 1, then for k = p - 1, the term 
(either n*2^k + 1 or 2^k + n ) is a multiple \
  of p (and also > p, so not prime).

to

%C A076336 Conjecturally, the numbers n such that n+2^k is composite for all 
k.

This adequately summarizes the discussion of S1 = A076336 and S2.
The discussion of S3 and S4 is irrelevant to A076336, and the
proof of their emptiness is simply a reworking of the well-known
proof that unbounded linear recurrences always include nonprimes.