A230194 - OEIS

%I #12 Aug 05 2019 05:32:32

%S 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,

%T 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,

%U 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

%N 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.

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

%C 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.

%C 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.

%H Zhi-Wei Sun, <a href="/A230194/b230194.txt">Table of n, a(n) for n = 1..5000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588 [math.NT], 2012-2017.

%e 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.

%t SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n-5]&&PrimeQ[6n+5]

%t 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}]

%t Table[a[n],{n,1,100}]

%Y Cf. A230140, A230141.

%K nonn

%O 1,8

%A _Zhi-Wei Sun_, Oct 11 2013