A232186 Number of ways to write n = p + q (q > 0) with p and p^3 + n*q^2 both prime.

0, 0, 1, 1, 2, 1, 1, 2, 3, 1, 1, 1, 2, 2, 3, 2, 2, 5, 1, 1, 3, 1, 5, 4, 2, 3, 3, 1, 2, 3, 2, 4, 6, 2, 3, 5, 2, 3, 3, 3, 2, 3, 4, 2, 4, 3, 2, 2, 3, 2, 6, 2, 3, 3, 5, 4, 4, 4, 5, 9, 1, 4, 7, 3, 4, 6, 3, 5, 8, 3, 5, 6, 5, 5, 13, 2, 4, 5, 4, 4, 7, 5, 5, 13, 3, 5, 8, 6, 4, 6, 4, 3, 8, 3, 4, 9, 1, 4, 11, 3

Offset: 1

Author

Zhi-Wei Sun, Nov 20 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 2.

Examples

			a(10) = 1 since 10 = 7 + 3 with 7 and 7^3 + 10*3^2 = 433 both prime.
a(11) = 1 since 11 = 5 + 6 with 5 and 5^3 + 11*6^2 = 521 both prime.
a(124) = 1 since 124 = 19 + 105 with 19 and 19^3 + 124*105^2 = 1373959 both prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588.

Crossrefs

Cf. A000040, A219864, A220413, A232174.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[Prime[k]^3+n*(n-Prime[k])^2],1,0],{k,1,PrimePi[n-1]}]
Table[a[n],{n,1,100}]