A388408 Numbers k such that 128 * 3^k + 1 is prime.

2, 3, 4, 8, 40, 170, 195, 215, 380, 495, 522, 562, 624, 650, 668, 804, 888, 978, 2830, 5319, 5903, 6479, 9650, 14510, 37838, 38807, 55982, 69827, 102010, 149308, 249792

Offset

1,1

Comments

a(32) > 254000.

Conjecture: This sequence intersects with A384228 at k = 2 to form twin primes with center N = 2^7 * 3^2 = 1152 = A027856(9). Any such intersection has to be at an odd k because if k is even, N-1 will be the difference of two squares, so not prime for N > 4. A covering system can be constructed that eliminates all other intersections (see linked program), and the search up to k = 254000 makes the probability of another intersection vanishingly small.

Links

Table of n, a(n) for n=1..31.

Ken Clements, Python program to calculate covering system.

Mathematica

Select[Range[0, 1000], PrimeQ[128*3^# + 1] &] (* Amiram Eldar, Sep 22 2025 *)

Prog

(Python)
from gmpy2 import is_prime
print([ k for k in range(4000) if is_prime(128 * 3**k + 1, 50)])

Crossrefs

Cf. A027856, A003306, A005537, A005538, A387060, A387201, A387761, A384228.

Sequence in context: A128657 A395753 A105421 * A273063 A217353 A258194

Adjacent sequences: A388405 A388406 A388407 * A388409 A388410 A388411

Keyword

nonn,more,hard

Author

Ken Clements, Sep 16 2025

Status

approved