A187758 Number of ways to write n=x+y (x,y>0) with 2x-3, 2x+3, 6y+1 and 6y+5 all prime

0, 0, 0, 0, 1, 2, 2, 2, 2, 3, 4, 2, 2, 3, 3, 3, 2, 3, 3, 4, 5, 3, 6, 5, 4, 6, 3, 5, 4, 3, 6, 2, 4, 5, 5, 4, 4, 6, 5, 4, 6, 5, 4, 5, 7, 5, 2, 3, 6, 4, 5, 4, 5, 7, 6, 9, 5, 4, 9, 5, 4, 5, 5, 4, 5, 6, 3, 8, 5, 8, 8, 3, 7, 5, 3, 5, 3, 5, 4, 9, 6, 4, 9, 7, 5, 8, 7, 8, 6, 9, 8, 2, 7, 7, 5, 6, 2, 10, 6, 3

Offset

1,6

Comments

Conjecture: a(n)>0 for all n>4.

This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes and also infinitely many sexy primes.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(5)=1 since 5=4+1 with 2*4-3, 2*4+3, 6*1+1 and 6*1+5 all prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[2k-3]==True&&PrimeQ[2k+3]==True&&PrimeQ[6(n-k)+1]==True&&PrimeQ[6(n-k)+5]==True, 1, 0], {k, 1, n-1}]
Do[Print[n, " ", a[n]], {n, 1, 100}]

Crossrefs

Cf. A023200, A046132, A023201, A002375, A187757, A199920, A219055, A218867, A220455.

Sequence in context: A219795 A082602 A216644 * A171931 A221530 A262944

Adjacent sequences: A187755 A187756 A187757 * A187759 A187760 A187761

Keyword

nonn

Author

Zhi-Wei Sun, Jan 03 2013

Status

approved