cp's OEIS Frontend

The even members of A005538, halved. - Jeppe Stig Nielsen, Oct 13 2019

All terms correspond to verified primes, that is, not merely probable primes.

a(14) > 2*10^5.

a(33) > 10^5.

Conjecture: The only intersection with A385115 is at k = 3 where 2^4 * 3^3 = 432 = A027856(8).

Idea: For odd k > 3, covering systems ensure mutual exclusion:

If k = 1, 9, 13, 19, 25, 31, 37, 39, 43, 49, 55 (mod 60), then 7 or 31 divides (16*3^k+1).

If k = 5, 7, 11, 17, 23, 27, 29, 35, 41, 47, 53, 57, 59 (mod 60), then 11 or 13 divides (16*3^k-1).

If k = 15, 21, 33, 45, 51 (mod 60), various primes including {11,31,43,109,277,433,...} ensure at least one of 16*3^k +- 1 is composite.

If k = 3 (mod 60) and k > 3, the probability of intersection becomes vanishingly small.

Only k = 3 escapes all divisibility conditions. Verified to k = 10^5.

a(23) > 10^5.

Conjecture: This sequence intersects with A387197 at k = 4 to form twin primes with center N = 2^5 * 3^4 = 2592 = A027856(10). Any such intersection has to be at an even k because if k is odd, either N-1 or N+1 has to be divisible by 5. A covering system can be constructed that eliminates all other intersections except where k = 4(mod 60), and for k > 4 with k = 4(mod 60), the search up to 10^5 makes the probability of another intersection in this residue class vanishingly small.