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

1,2

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.

Links

Table of n, a(n) for n=1..12.

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

Example

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

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

Prog

(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

Crossrefs

Cf. A001359, A224695, A292691.

Sequence in context: A053725 A053774 A254570 * A009723 A069992 A012196

Adjacent sequences: A308401 A308402 A308403 * A308405 A308406 A308407

Keyword

nonn

Author

Amiram Eldar, Aug 04 2019

Status

approved