A387060 Numbers k such that 16 * 3^k + 1 is prime.

0, 3, 4, 5, 12, 24, 36, 77, 195, 296, 297, 533, 545, 644, 884, 932, 1409, 2061, 2453, 2985, 3381, 4980, 5393, 11733, 13631, 14516, 21004, 27663, 32645, 39453, 67055, 90543

Offset: 1

Author

Ken Clements, Aug 15 2025

Keywords

• nonn
• more
• new

Comments

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.

Links

• Ken Clements, Python program to calculate covering system.

Crossrefs

Cf. A027856, A005537, A005538, A385115.

Programs

Mathematica

Select[Range[0, 4000], PrimeQ[16*3^# + 1] &] (* Amiram Eldar, Aug 16 2025 *)

Python

from gmpy2 import is_prime
print([k for k in range(4_000) if is_prime(16 * 3**k + 1)])