A387060 -id:A387060 - cp's OEIS frontend

A385115 Numbers k such that 2^4 * 3^k - 1 is prime.

1, 3, 9, 13, 31, 43, 81, 121, 235, 1135, 1245, 1521, 2019, 2329, 3573, 11245, 15571, 37333, 54471, 70641

Offset: 1

Ken Clements, Aug 14 2025

All terms are odd, since if k were even, N = 2^4 * 3^k would be a perfect square and N - 1 could be factored as the difference of squares, hence not prime.

a(21) > 10^5. - Michael S. Branicky, Aug 15 2025

Cf. A005540, A005541, A387060.

Select[Range[4000], PrimeQ[16 * 3^# - 1] &] (* Amiram Eldar, Aug 15 2025 *)

from gmpy2 import is_prime
print([k for k in range(1, 4_000, 2) if is_prime(16 * 3**k - 1)])

a(17)-a(20) from Michael S. Branicky, Aug 15 2025

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

Original entry on oeis.org

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

Offset: 1

Ken Clements, Aug 21 2025

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.

Ken Clements, Python program to calculate covering system.

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

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

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

cp's OEIS Frontend

A385115 Numbers k such that 2^4 * 3^k - 1 is prime.

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