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%I #17 Oct 24 2013 02:50:18
%S 0,0,1,1,1,1,1,2,2,1,2,3,4,2,3,3,3,3,3,2,3,3,5,4,2,2,5,5,3,3,6,7,8,4,
%T 3,7,8,6,5,6,8,9,7,4,5,8,8,7,4,5,10,9,5,4,7,8,9,6,4,8,11,7,4,5,6,10,7,
%U 2,5,8,7,5,3,3,8,8,2,3,6,4,6,3,1,5,6,3,2,3,3,7,3,1,5,5,2,4,4,4,7,5
%N Number of ways to write n = a + b + c with a <= b <= c, where a, b, c are among those numbers m (terms of A230506) with 2*m + 1 and 2*m^3 + 1 both prime.
%C Conjecture: (i) a(n) > 0 for all n > 2.
%C (ii) Any integer n > 8 can be written as x + y + z (x, y, z > 0) with 2*x + 1, 2*y + 1, 2*z - 1, 2*x^4 - 1, 2*y^4 - 1, 2*z^4 - 1 all prime.
%C Either of the two parts of the conjecture is stronger than Goldbach's weak conjecture which was finally proved by H. Helfgott in 2013.
%C Part (i) implies that there are infinitely many positive integers n with 2*n + 1 and 2*n^3 + 1 both prime, and part (ii) implies that there are infinitely many positive integers n with 2*n + 1 and 2*n^4 - 1 both prime.
%C We have verified the conjecture for n up to 10^6.
%H Zhi-Wei Sun, <a href="/A230507/b230507.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;37a4cf5.1310">On representations via sparse primes</a>, a message to Number Theory List, Oct. 23, 2013.
%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(8) = 2 since 8 = 1 + 1 + 6 = 1 + 2 + 5, and 2*1 + 1 = 3, 2*1^3 + 1 = 3, 2*6 + 1 = 13, 2*6^3 + 1 = 433, 2*2 + 1 = 5, 2*2^3 + 1 = 17, 2*5 + 1 = 11, 2*5^3 + 1 = 251 are all prime.
%t pp[n_]:=PrimeQ[2n+1]&&PrimeQ[2n^3+1]
%t a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j],1,0],{i,1,n/3},{j,i,(n-i)/2}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A068307, A230219, A230351, A230493, A230502, A230506.
%K nonn
%O 1,8
%A _Zhi-Wei Sun_, Oct 21 2013
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