A282136 Numbers m such that both 2^k*3^(m-k) - 1 and 2^(m-k)*3^k - 1 are primes for some positive k < m.

2, 3, 4, 5, 7, 8, 9, 11, 13, 17, 20, 23, 25, 31, 43, 47, 85, 101, 117, 173, 224, 277, 281, 349, 359, 365, 403, 521, 629, 691, 709, 819, 1037, 1323, 1516, 1621, 1807, 1928, 3007, 3031, 3513

Offset

1,1

Comments

a(42) > 7000. - Michael S. Branicky, May 06 2025

Links

Table of n, a(n) for n=1..41.

Robert de Mello Koch, Animik Ghosh, and Hendrik J. R. Van Zyl, Bosonic Fortuity in Vector Models, arXiv:2504.14181 [hep-th], 2025. See p. 10.

Example

2 is in this sequence because 2^1*3^(2-1) - 1 = 5 is prime and 2^(2-1)*3^1 - 1 = 5 is prime.
3 is in this sequence because 2^1*3^(3-1) - 1 = 17 is prime and 2^(3-1)*3^1 - 1 = 11 is prime.
4 is in this sequence because 2^1*3^(4-1) - 1 = 53 is prime and 2^(4-1)*3^1 - 1 = 23 is prime.
5 is in this sequence because 2^2*3^(5-2) - 1 = 107 is prime and 2^(5-2)*3^2 - 1 = 71 is prime.

Mathematica

Select[Range@ 800, Function[n, Total@ Boole@ Table[PrimeQ@ {2^k*3^(n - k) - 1 , 2^(n - k)*3^k - 1} == {True, True}, {k, n/2}] > 0]] (* Michael De Vlieger, Feb 07 2017 *)

Prog

(PARI) is(n)=for(k=1, n-1, if(isprime(3^(n-k)<<k - 1) && isprime(3^k<<(n-k) - 1), return(1))); 0 \\ Charles R Greathouse IV, Feb 21 2017

Crossrefs

Sequence in context: A087797 A340815 A280619 * A371812 A153730 A140691

Adjacent sequences: A282133 A282134 A282135 * A282137 A282138 A282139

Keyword

nonn,more

Author

Juri-Stepan Gerasimov, Feb 06 2017

Extensions

a(21)-a(33) from Michael De Vlieger, Feb 07 2017

a(34)-a(41) from Michael S. Branicky, May 05 2025

Status

approved