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%I #12 Mar 30 2014 19:32:21
%S 0,1,1,2,2,0,2,3,2,3,2,3,4,1,2,1,4,3,1,1,6,4,2,4,4,4,1,5,3,5,6,4,6,3,
%T 5,5,6,3,3,5,5,5,9,3,3,11,6,7,4,8,7,12,7,5,10,4,3,8,8,3,11,6,5,10,4,6,
%U 14,6,3,9,3,12,12,9,3,11,6,10,15,7,7,8,3,6,11,8,11,10,7,3,11,10,7,11,4,6,13,11,9,8
%N Number of ways to write n = x + y (x, y > 0) with x^4 + y*(y+1)/2 prime.
%C Conjecture: (i) a(n) > 0 except for n = 1, 6.
%C (ii) For any positive integer n not among 1, 3, 14, 25, there are positive integers x and y with x + y = n such that x^3 + y*(y+1)/2 is prime.
%H Zhi-Wei Sun, <a href="/A228430/b228430.txt">Table of n, a(n) for n = 1..10000</a>
%e a(14) = 1 since 14 = 4 + 10 with 4^4 + 10*11/2 = 311 prime.
%e a(27) = 1 since 27 = 22 + 5 with 22^4 + 5*6/2 = 234271 prime.
%t a[n_]:=Sum[If[PrimeQ[x^4+(n-x)(n-x+1)/2],1,0],{x,1,n-1}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000217, A000583, A228425, A228428, A228429.
%K nonn
%O 1,4
%A _Zhi-Wei Sun_, Nov 10 2013
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