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%I #9 Jun 03 2018 11:08:23
%S 0,1,1,1,1,1,1,1,2,1,1,1,4,2,3,1,4,2,2,1,4,1,6,2,5,2,7,1,3,2,6,2,7,2,
%T 2,4,3,3,3,3,6,1,7,3,6,7,6,3,4,1,6,2,8,3,5,3,7,5,8,4,5,3,7,4,6,4,7,5,
%U 7,5,11,6,8,4,9,3,8,4,6,5,9,5,11,6,6,6,16,7,10,5
%N Number of ways to write n as x + y with 0 < x <= y such that x^3 + n*y^2 is prime.
%C Conjecture: a(n) > 0 for all n > 1, and a(n) = 1 only for n = 2, 3, 4, 5, 6, 7, 8, 10, 11, 12, 16, 20, 22, 28, 42, 50.
%C We have verified a(n) > 0 for all n = 2..10^7.
%C See also A232174 for a similar conjecture.
%H Zhi-Wei Sun, <a href="/A305502/b305502.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="https://doi.org/10.1007/978-3-319-68032-3_20">Conjectures on representations involving primes</a>, in: M. Nathanson (ed.), Combinatorial and Additive Number Theory II, Springer Proc. in Math. & Stat., Vol. 220, Springer, Cham, 2017, pp. 279-310. (See also <a href="http://arxiv.org/abs/1211.1588">arXiv:1211.1588 [math.NT]</a>, 2012-2017.)
%e a(2) = 1 since 2 = 1 + 1 with 1^3 + 2*1^2 = 3 prime.
%e a(3) = 1 since 3 = 1 + 2 with 1^3 + 3*2^2 = 13 prime.
%e a(11) = 1 since 11 = 5 + 6 with 5^3 + 11*6^2 = 521 prime.
%e a(20) = 1 since 20 = 3 + 17 with 3^3 + 20*17^2 = 5807 prime.
%e a(28) = 1 since 28 = 9 + 19 with 9^3 + 28*19^2 = 10837 prime.
%e a(42) = 1 since 42 = 19 + 23 with 19^3 + 42*23^2 = 29077 prime.
%e a(50) = 1 since 50 = 3 + 47 with 3^3 + 50*47^2 = 110477 prime.
%t tab={};Do[r=0;Do[If[PrimeQ[x^3+n(n-x)^2],r=r+1],{x,1,n/2}];tab=Append[tab,r],{n,1,90}];Print[tab]
%Y Cf. A000040, A000290, A000578, A232174.
%K nonn
%O 1,9
%A _Zhi-Wei Sun_, Jun 03 2018
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