I remember reading some passing comments on the apparent 'weights' of n's in files with many k's, (like NPLB's

drive 11, or

drive 1, where I think it was first noticed) and that we chalked it up to random chance. But then I looked at some (e.g.

k*2^333333-1) with factordb.com, and it looks pretty obvious to me that, just like with fixed k's, (e.g.

349*2^n-1) k*2^n-1 with fixed n has factors repeating periodically (just like with fixed k, this can mean that the number of candidates remaining after trivial factorization can be a small or large proportion of the candidates).

I wonder: can there be the fixed-n equivalents of Riesel/Sierpinski numbers? (i.e. numbers n where all numbers k*2^n-1 with k>0, n>0 are composite) If so, what is the smallest such? And is this mathematically any more or less interesting than Riesel/Sierpinski numbers?

I can see that, for practical purposes of proving such a conjecture, it'd be easier. Because you can test near-countless candidates before the digit length of the candidates you're testing grows much larger than the original 1*2^n-1.

(I figured CRUS would be the best subforum for this, since it's not serious enough for the likes of the Math forum, and is more CRUS-like with the vague consideration of covering sets, etc. than NPLB)