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%I #17 Oct 25 2013 14:40:04
%S 0,0,1,0,2,3,0,4,3,1,7,3,2,3,7,4,5,6,3,4,8,5,8,3,6,8,9,9,5,12,2,11,4,
%T 4,4,13,5,9,13,8,14,8,3,15,7,8,10,6,5,17,15,4,6,9,8,10,15,9,7,15,11,5,
%U 6,11,14,14,7,11,3,12,23,16,5,20,14,4,9,14,5,19,19,4,3,12,7,16,5,12,6,11,12,12,23,14,23,12,5,17,14,5
%N Number of ways to write n = x + y + z (x, y, z > 0) such that 2*x + 1, 2*y + 3, 2*z + 5 are all prime and x*y*z is a triangular number.
%C Conjecture: (i) a(n) > 0 except for n = 1, 2, 4, 7.
%C (ii) Any integer n > 7 can be written as x + y + z (x, y, z > 0) such that 2*x + 1, 2*y + 1, 2*x*y + 1 are primes and x*y*z is a triangular number.
%C (iii) Each integer n > 4 not equal to 7 or 14 can be expressed as p + q + r (p, q, r > 0) with p and 2*q + 1 both primes, and p*q*r a triangular number.
%C (iv) Any integer n > 6 not among 16, 20, 60 can be written as x + y + z (x, y, z > 0) such that x*y + x*z + y*z is a triangular number.
%C Part (i) is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013.
%C See also A227877 and A230596 for some related conjectures.
%H Zhi-Wei Sun, <a href="/A230451/b230451.txt">Table of n, a(n) for n = 1..6000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;4704f25.1310">A new conjecture on triangular numbers</a>, a message to Number Theory List, Oct. 25, 2013.
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.
%e a(6) = 3 since 6 = 1 + 2 + 3 = 2 + 1 + 3 = 3 + 2 + 1, and 2*1 + 1 = 3, 2*2 + 3 = 7, 2*3 + 5 = 11, 2*2 + 1 = 5, 2*1 + 3 = 5, 2*3 + 1 = 7, 2*1 + 5 = 7 are all prime.
%e a(10) = 1 since 10 = 3 + 4 + 3, and 2*3 + 1 = 7, 2*4 + 3 = 11, 2*3 + 5 = 11 are all prime.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t TQ[n_]:=SQ[8n+1]
%t a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[2j+3]&&PrimeQ[2(n-i-j)+5]&&TQ[i*j(n-i-j)],1,0],{i,1,n-2},{j,1,n-1-i}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000217, A068307, A132399, A227877, A229166, A230121, A230141, A230219, A230596.
%K nonn
%O 1,5
%A _Zhi-Wei Sun_, Oct 19 2013
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