A384228 Numbers k such that 128 * 3^k - 1 is prime.

0, 1, 2, 9, 13, 25, 32, 36, 53, 104, 146, 212, 213, 305, 396, 636, 901, 4050, 4778, 4881, 4982, 5084, 8821, 10649, 11685, 13028, 16340, 29096, 44480, 56628, 66441, 99698, 104966, 109538, 114216, 135717, 140229

Offset

1,3

Comments

a(38) > 250000.

Conjecture: This sequence intersects with A388408 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 = 250000 makes the probability of another intersection vanishingly small.

Links

Table of n, a(n) for n=1..37.

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, A003307, A005540, A005541, A385115, A387197, A387760, A388408.

Sequence in context: A115907 A063634 A396189 * A178688 A163735 A144472

Adjacent sequences: A384225 A384226 A384227 * A384229 A384230 A384231

Keyword

nonn,more,hard

Author

Ken Clements, Sep 16 2025

Status

approved