cp's OEIS Frontend

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.

%I A230141 #18 Jan 08 2025 10:45:18
%S A230141 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,
%T A230141 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,
%U A230141 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
%N 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.
%C A230141 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.
%C A230141 This is a further refinement of the conjecture in A230140.
%C A230141 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.
%H A230141 Zhi-Wei Sun, <a href="/A230141/b230141.txt">Table of n, a(n) for n = 1..10000</a>
%H A230141 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.
%e A230141 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.
%t A230141 SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[12n-7]
%t A230141 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}]
%t A230141 Table[a[n],{n,1,100}]
%Y A230141 Cf.  A001359, A006512, A002375, A068307, A230138, A230140.
%K A230141 nonn
%O A230141 1,4
%A A230141 _Zhi-Wei Sun_, Oct 10 2013