A258139 Number of ways to write n as p^2 + q with p and q both prime.

0, 0, 0, 0, 0, 1, 1, 0, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 1, 2, 2, 0, 1, 0, 2, 1, 0, 1, 1, 0, 2, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 2, 2, 0, 2, 0, 3, 1, 0, 0, 1, 0, 3, 1, 0, 1, 2, 0, 3, 0, 1, 1, 2, 0, 0, 1, 1, 1, 2, 0, 2, 0, 1, 1, 1, 0, 2, 1, 1, 0, 1, 0, 3, 1, 0, 0, 2, 0, 2, 0, 0

Offset: 1

Author

Zhi-Wei Sun, May 22 2015

Keywords

• nonn

Comments

Conjecture: For any integer n > 0, we have a(n+r) > 0 for some r = 0,1,2,3,4,5. Moreover, if n = 6*k + 2, then a(n) > 0 except for k = 0, 1, 12, 28, 102, 117, 168, 4079.

We have verified the conjecture for n up to 10^9.

Examples

			a(11) = 2 since 11 = 2^2 + 7 = 3^2 + 2 with 2, 3, 7 all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000040, A002375, A258140, A258141.

Programs

Mathematica

Do[r=0;Do[If[PrimeQ[n-Prime[k]^2],r=r+1],{k,1,PrimePi[Sqrt[n]]}];Print[n," ",r];Continue,{n,1,100}]

Formula

G.f.: (Sum_{k>=1} x^prime(k))*(Sum_{k>=1} x^(prime(k)^2)). - Ilya Gutkovskiy, Feb 05 2017