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%I #15 Sep 14 2014 21:26:34
%S 0,0,1,1,2,2,3,2,3,1,2,2,2,3,3,4,2,3,2,3,3,3,4,2,5,2,6,3,6,5,4,5,3,5,
%T 5,8,7,6,5,6,5,5,7,6,8,4,6,5,6,7,9,8,8,5,7,6,8,10,6,10,4,8,6,6,10,6,9,
%U 5,6,5,7,7,9,6,7,8,5,10,6,9,6,6,7,4,7,7,9,6,5,5,4,6,5,6,5,5,6,4,6,6
%N Number of ways to write n = x + y + z with 0 < x <= y <= z such that 6*x-1, 6*y-1, 6*z-1 are among those primes p (terms of A230138) with p + 2 and 2*p - 5 also prime.
%C Conjecture: (i) a(n) > 0 for all n > 2, i.e., 6*n-3 with n > 2 can be expressed as a sum of three terms from A230138. Moreover, for any integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x-1, 6*y-1, 6*z-1 are primes in A230138.
%C (ii) For each integer n > 12, there are three distinct positive integers x, y, z with x + y + z = n such that 6*x-1, 6*y-1, 6*z-1 are among those primes p with p + 2 and 2*p + 9 also prime.
%C Note that part (i) of this conjecture implies that there are infinitely many primes in A230138.
%C Indices k such that a(m)>a(k) for all m>k, are (2, 10, 26, 334, 439, 544, 551, 684, ...). The only sequence which has the first 5 terms within the 3 lines of data is A212067. (Certainly a coincidence.) - _M. F. Hasler_, Oct 10 2013
%H Zhi-Wei Sun, <a href="/A230140/b230140.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.
%e a(10) = 1 since 10 = 2 + 3 + 5, and the three numbers 6*2-1=11, 6*3-1=17 and 6*5-1=29 are terms of A230138.
%t SQ[n_]:=PrimeQ[6n-1]&&PrimeQ[6n+1]&&PrimeQ[12n-7]
%t a[n_]:=Sum[If[SQ[i]&&SQ[j]&&SQ[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
%t Table[a[n],{n,1,100}]
%o (PARI) ip(x)=isprime(6*x-1) && isprime(6*x+1) && isprime(12*x-7); a(n)=sum(x=1,n\3,sum(y=x,ip(x)*(n-x)\2,ip(y) && ip(n-x-y))) \\ - _M. F. Hasler_, Oct 10 2013
%Y Cf. A068307, A230138, A230141.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Oct 10 2013
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