A230194 Number of ways to write n = x + y + z (x, y, z > 0) such that all the 11 integers 6*x-1, 6*x+1, 6*x-5, 6*x+5, 6*y-1, 6*y-5, 6*y+5, 6*(x+y)+5, 6*z-1, 6*z-5 and 6*z+5 are prime.

0, 0, 0, 0, 0, 1, 1, 3, 3, 3, 4, 3, 6, 5, 3, 3, 3, 5, 4, 4, 4, 2, 9, 10, 9, 7, 5, 12, 8, 2, 8, 6, 6, 7, 9, 4, 3, 10, 11, 2, 4, 6, 10, 9, 11, 9, 4, 10, 17, 9, 1, 4, 7, 6, 6, 6, 2, 5, 14, 13, 7, 5, 14, 6, 3, 5, 4, 12, 11, 14, 5, 2, 16, 11, 5, 9, 6, 8, 11, 23, 15, 3, 23, 18, 17, 9, 8, 20, 5, 10, 14, 3, 14, 15, 16, 9, 8, 24, 10, 7

Offset

1,8

Comments

Conjecture: a(n) > 0 for all n > 5.

Let S be the set of those primes p with p-4 and p+6 also prime. Since each element of S has the form 6*k-1 with k > 0, the conjecture implies that 6*n-3 with n > 5 can be expressed as a sum of three primes in the set S. If n = x + y + z, then 6*n = (6*(x+y)+5) + (6*z-5). So a(n) > 0 implies Goldbach's conjecture for the even number 6*n.

Let T be the set of those primes p with p+4 and p-6 also prime. Clearly each element of T has the form 6*k+1 with k > 0. We conjecture that 6*n+3 with n > 5 can be expressed as a sum of three primes in the set T.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

Example

a(30) = 2 since 30 = 2 + 14 + 14 = 18 + 4 + 8, and 6*2-1 = 11, 6*2+1 = 13, 6*2-5 = 7, 6*2+5 = 17, 6*14-1 = 83, 6*14-5 = 79, 6*14+5 = 89, 6*(2+14)+5 = 101, 6*18-1 = 107, 6*18+1 = 109, 6*18-5 = 103, 6*18+5 = 113, 6*4-1 = 23, 6*4-5 = 19, 6*4+5 = 29, 6*(18+4)+5 = 137, 6*8-1 = 47, 6*8-5 = 43 and 6*8+5 = 53 are all prime.

Mathematica

SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n-5]&&PrimeQ[6n+5]
a[n_]:=Sum[If[SQ[i]&&PrimeQ[6i+1]&&SQ[j]&&PrimeQ[6(i+j)+5]&&SQ[n-i-j], 1, 0], {i, 1, n-2}, {j, 1, n-1-i}]
Table[a[n], {n, 1, 100}]

Crossrefs

Cf. A230140, A230141.

Sequence in context: A098037 A079108 A165605 * A128112 A122669 A071942

Adjacent sequences: A230191 A230192 A230193 * A230195 A230196 A230197

Keyword

nonn

Author

Zhi-Wei Sun, Oct 11 2013

Status

approved