A387201 Numbers k such that 32 * 3^k + 1 is prime.

1, 4, 8, 9, 32, 36, 48, 74, 112, 186, 204, 364, 393, 572, 781, 1208, 2624, 2778, 4522, 4896, 5272, 32884

Offset: 1

Author

Ken Clements, Aug 21 2025

Keywords

• nonn
• more
• new

Comments

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.

Links

• Ken Clements, Python program to calculate covering system.

Crossrefs

Cf. A027856, A003306, A005537, A005538, A387060, A387197.

Programs

Mathematica

Select[Range[0, 5000], PrimeQ[32 * 3^# + 1] &] (* Amiram Eldar, Aug 21 2025 *)

Python

from gmpy2 import is_prime
print([ k for k in range(4000) if is_prime(32 * 3**k + 1)])