A308404 a(n) = Q(A001359(n)), with Q(n) = (2^(n+2)-3n-8)/(n*(n+2)).

1, 3, 57, 1623, 2388747, 4989275679, 640689916425033, 1822252163947383837, 974834644028245238101857699, 55649241817444349958527998041, 36596034629737014817675324057147576383, 126872100333877939558050221738699065414707

Offset: 1

Author

Amiram Eldar, Aug 04 2019

Keywords

• nonn

Comments

Aebi and Cairns proved that if (p, p+2) are twin primes, then 2^(p+2) == 3p+8 (mod p(p+2)). This sequence contains the quotients of this congruence.

Examples

			a(2) = 3, because A001359(2) = 5 and Q(5) = (2^(5+2)-3*5-8)/(5*(5+2)) = 3.
		

Links

• Christian Aebi and Grant Cairns, Catalan numbers, primes, and twin primes, Elemente der Mathematik, Vol. 63, No. 4 (2008), pp. 153-164.

Crossrefs

Cf. A001359, A224695, A292691.

Programs

Mathematica

s={}; Do[If[PrimeQ[n] && PrimeQ[n+2], q = (2^(n+2)-3n-8)/(n(n+2)); AppendTo[s,q]], {n, 1, 1000}]; s

PARI

lista(nn) = {forprime(p=2, nn, if (isprime(p+2), print1((2^(p+2)-3*p-8)/(p*(p+2)), ", ");););} \\ Michel Marcus, Aug 04 2019