A187759 Number of ways to write n=x+y (0<x<y<n) with 6x-1, 6x+1, 6y-1 and 6y+1 all prime.

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 2, 0, 2, 1, 3, 2, 1, 2, 1, 2, 2, 2, 2, 3, 1, 3, 1, 2, 3, 2, 6, 1, 3, 1, 2, 4, 3, 4, 4, 1, 3, 1, 3, 5, 2, 6, 1, 3, 2, 2, 5, 2, 5, 2, 3, 1, 2, 3, 5, 2, 4, 0, 0, 3, 1, 6, 2, 3, 3, 1, 5, 1, 5, 3, 3, 3, 1, 4, 2, 3, 3, 0, 3, 3, 3, 4, 1, 3, 1, 2, 3, 2, 4, 2, 2, 3

Offset

1,8

Comments

Conjecture: If n>200 is not among 211, 226, 541, 701, then a(n)>0.

This essentially follows from the conjecture related to A219157, since n=x+y for some positive integers x and y with 6x-1,6x+1,6y-1,6y+1 all prime if and only if 6n=p+q for some twin prime pairs {p,p-2} and {q,q+2}.

Similarly, the conjecture related to A218867 implies that any integer n>491 can be written as x+y (0<x<=y<n) with 6x+1, 6x+5, 6y+1 and 6y+5 all prime; and the conjecture related to A219055 implies that any integer n>1600 not among 2729 and 4006 can be written as x+y (0<x<=y<n) with 2x-3, 2x+3, 2y-3 and 2y+3 all prime.

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(9)=1 since 9=2+7 with 6*2-1, 6*2+1, 6*7-1 and 6*7+1 all prime.

Mathematica

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

Prog

(PARI) a(n)=sum(x=1, (n-1)\2, isprime(6*x-1)&&isprime(6*x+1)&&isprime(6*n-6*x-1)&&isprime(6*n-6*x+1)) \\ Charles R Greathouse IV, Feb 28 2013

Crossrefs

Cf. A001359, A006512, A219157, A219185, A199920, A187757, A187758, A218867, A219055.

Sequence in context: A242748 A266012 A202111 * A358565 A270645 A350239

Adjacent sequences: A187756 A187757 A187758 * A187760 A187761 A187762

Keyword

nonn

Author

Zhi-Wei Sun, Jan 03 2013

Status

approved