A230141 Number of ways to write n = x + y + z with y <= z such that 6*x-1, 6*y-1, 6*z-1 are terms of A230138 and 6*(y+z)+1 is prime.

0, 0, 1, 2, 2, 2, 4, 5, 3, 2, 3, 4, 4, 5, 6, 5, 3, 5, 4, 4, 2, 4, 6, 2, 3, 2, 6, 9, 8, 8, 5, 5, 4, 5, 10, 14, 10, 12, 6, 11, 7, 9, 13, 6, 11, 3, 9, 7, 8, 14, 6, 11, 4, 4, 8, 9, 15, 15, 7, 14, 3, 6, 13, 10, 19, 6, 6, 12, 5, 10, 8, 7, 16, 6, 10, 4, 7, 19, 11, 13, 3, 12, 5, 6, 13, 5, 12, 7, 8, 4, 5, 6, 10, 6, 4, 6, 4, 6, 7, 7

Offset: 1

Author

Zhi-Wei Sun, Oct 10 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 2. Also, any integer n > 2 can be written as x + y + z (x, y, z > 0) such that 6*x-1, 6*y-1, 6*z-1 are terms of A230138 and 6*y*z-1 is prime.

This is a further refinement of the conjecture in A230140.

Note that if x + y + z = n then 6*n = (6*x-1) + (6*(y+z)+1). So a(n) > 0 implies Goldbach's conjecture for the even number 6*n.

Examples

			a(10) = 2 since 10 = 3 + 2 + 5 = 5 + 2 + 3, and 6*3-1 = 17, 6*2-1 = 11, 6*5-1 = 29 are terms of A230138, and 6*(2+5)+1 = 43 and 6*(2+3)+1 = 31 are also prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

Cf. A001359, A006512, A002375, A068307, A230138, A230140.

Programs

Mathematica

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[12n-7]
a[n_]:=Sum[If[SQ[i]&&PrimeQ[6(n-i)+1]&&SQ[j]&&SQ[n-i-j],1,0],{i,1,n-2},{j,1,(n-i)/2}]
Table[a[n],{n,1,100}]