A386857 Numbers k such that both 9*2^k - 1 and 9*2^k + 1 are prime.

1, 3, 7, 43, 63, 211

Offset: 1

Author

Ken Clements, Aug 05 2025

Keywords

• nonn
• more

Comments

The exponent, k, of 2 must be odd because the exponent, 2, of 3 (where 9 = 3^2) is even and the sum of the exponents of prime factors 2 and 3 must be odd to form a product that is a twin prime average. Of all subsequences of A027856, this is the longest known where the power of 3 is fixed.

Amiram Eldar noted that using A002236 and A002256, we obtain a(7) > 5.6*10^6, if it exists.

Examples

			a(1) = 1 because 2*9 = 18 with 17 and 19 prime.
a(2) = 3 because 8*9 = 72 with 71 and 73 prime.
a(3) = 7 because 128*9 = 1152 with 1151 and 1153 prime.
a(4) = 43 because 8796093022208*9 = 79164837199872 with 79164837199871 and 79164837199873 prime.
		

Crossrefs

Intersection of A002236 and A002256.

Cf. A014574, A384530, A027856, A385433, A181490.

Programs

Maple

q:= k-> (m-> andmap(isprime, [m-1, m+1]))(9*2^k):
select(q, [2*i-1$i=1..111])[];  # Alois P. Heinz, Aug 08 2025

Python

from gmpy2 import is_prime
def is_TPpi2(e2, e3):
    N = 2**e2 * 3**e3
    return is_prime(N-1) and is_prime(N+1)
print([k for k in range(1, 100001, 2) if is_TPpi2(k, 2)])