A387197 Numbers k such that 32 * 3^k - 1 is prime.

0, 3, 4, 6, 46, 59, 84, 94, 124, 239, 267, 366, 371, 424, 616, 2139, 2299, 3523, 3563, 3843, 3923, 7627, 12751, 34798, 39911, 56568, 58779

Offset: 1

Author

Ken Clements, Aug 21 2025

Keywords

• nonn
• more
• new

Comments

a(28) > 10^5.

Conjecture: This sequence intersects with A387201 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, A003307, A005540, A005541, A385115, A387201.

Programs

Mathematica

Select[Range[0, 4000], 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)])