A332847 a(n) is the smallest k such that exactly one of k*2^(2^n) - 2*k + 1 and k*2^(2^n) + 2*k - 1 is a prime.

1, 4, 1, 1, 1, 3, 3, 10, 17, 8, 83, 92, 525, 1888, 20, 6804, 11390

Offset: 0

Author

Juri-Stepan Gerasimov, Feb 26 2020

Keywords

• nonn
• more

Comments

Conjecture: a(n) > 1 for all n > 4.

a(n) = 1, n > 1, is equivalent to F(n) = 2^(2^n) + 1 being a Fermat prime, because the Mersenne number M(2^n) = 2^(2^n) - 1 is prime only for n = 1 (since divisible by 3 for all n >= 1), where F(1) is also prime. - The two considered numbers can also be written 2*k*A077585(n) + 1 resp. 2*k*A000051(A000225(n)) - 1. - M. F. Hasler, Mar 05 2020

Examples

			a(0) = 1 because 1*2^(2^0) - 2*1 + 1 = 1 is a nonprime and 1*2^(2^0) + 2*1 - 1 = 3 is a prime.
a(1) = 4 because 4*2^(2^1) - 2*4 + 1 = 9 is a composite and 4*2^(2^1) + 2*4 - 1 = 23 is a prime.
a(2) = 1 because 1*2^(2^2) - 2*1 + 1 = 15 is a composite and 1*2^(2^2) + 2*1 - 1 = 17 is a prime.
a(3) = 1 because 1*2^(2^3) - 2*1 + 1 = 255 is a composite and 1*2^(2^3) + 2*1 - 1 = 257 is a prime.
a(4) = 1 because 1*2^(2^4) - 2*1 + 1 = 65535 is a composite and 1*2^(2^4) + 2*1 - 1 = 65537 is a prime.
		

Crossrefs

Cf. A019434, A331487.

Cf. A000215 (Fermat numbers 2^2^n + 1), A000225 (Mersenne numbers 2^n - 1).

Programs

Mathematica

Table[Module[{k=1},While[Total[Boole[PrimeQ[k*2^(2^n)+{2k-1,-2k+1}]]]!=1,k++];k],{n,0,14}] (* Harvey P. Dale, Jun 03 2025 *)

PARI

a(n) = {my(k=1, m=2^2^n); while(ispseudoprime(k*m-2*k+1)-ispseudoprime(k*m+2*k-1)==0, k++); k; } \\ Jinyuan Wang, Feb 26 2020

Extensions

Offset changed to 0 and a(11)-a(14) from Jinyuan Wang, Feb 26 2020

a(15)-a(16) from Michael S. Branicky, Jun 16 2025