Quote:
Originally Posted by sweety439
9*1024^n1 is unlikely to have a covering set, like the simple cases: 3*2^n+1, 5*2^n+1, 7*2^n+1, 9*2^n+1, 11*2^n+1, etc. they are unlikely to have a covering set, but also no proof.

Well... since all prime factors of k*2^n+1 are odd, thus if there exist an n such that k*2^n+1 or its dual (2^n+k) is power of 2 (including 1), then k*2^n+1 cannot have a covering set.
Thus these k*2^n+1 cannot have a covering set:
1*2^n+1 (for n=0, the value is 2)
3*2^n+1 (for n=0, the value is 4)
7*2^n+1 (for n=0, the value is 8)
15*2^n+1 (for n=0, the value is 16)
31*2^n+1 (for n=0, the values is 32)
1*2^n1 (the dual form 2^n1, for n=1, the value is 1)
3*2^n1 (for n=0, the value is 2, also the dual form 2^n3, for n=2, the value is 1)
5*2^n1 (for n=0, the value is 4)
7*2^n1 (the dual form 2^n7, for n=3, the value is 1)
9*2^n1 (for n=0, the value is 8)
15*2^n1 (the dual form 2^n15, for n=4, the value is 1)
17*2^n1 (for n=0, the value is 16)
31*2^n1 (the dual form 2^n31, for n=5, the value is 1)
33*2^n1 (for n=0, the value is 32)
etc.
However, there is no proof that 5*2^n+1, 9*2^n+1, 11*2^n+1, 13*2^n+1, 17*2^n+1, ... have no covering set (i.e. the sequence of the smallest prime factor of k*2^n+1 is unbounded above).