A230252 Number of ways to write n = x + y (x, y > 0) with 2*x + 1, x^2 + x + 1 and y^2 + y + 1 all prime.

0, 1, 2, 3, 2, 3, 4, 4, 4, 3, 4, 1, 3, 3, 3, 5, 5, 4, 3, 6, 4, 7, 7, 2, 4, 6, 4, 4, 6, 3, 1, 4, 2, 4, 7, 4, 1, 4, 4, 2, 6, 4, 3, 4, 2, 3, 5, 3, 2, 1, 2, 3, 6, 2, 6, 6, 3, 5, 4, 5, 3, 7, 2, 4, 6, 2, 4, 5, 3, 5, 8, 5, 2, 10, 4, 4, 8, 5, 6, 7, 8, 4, 11, 4, 3, 6, 4, 2, 4, 8, 8, 11, 5, 3, 11, 5, 3, 6, 4, 5

Offset: 1

Author

Zhi-Wei Sun, Oct 13 2013

Keywords

• nonn

Comments

Conjecture: (i) a(n) > 0 for all n > 1.

(ii) Any integer n > 3 can be written as p + q with p, 2*p - 3 and q^2 + q + 1 all prime. Also, each integer n > 3 not equal to 30 can be expressed as p + q with p, q^2 + q - 1 and q^2 + q + 1 all prime.

(iii) Any integer n > 1 can be written as x + y (x, y > 0) with x^2 + 1 (or 4*x^2+1) and y^2 + y + 1 (or 4*y^2 + 1) both prime.

(iv) Each integer n > 3 can be expressed as p + q (q > 0) with p, 2*p - 3 and 4*q^2 + 1 all prime.

(v) Any even number greater than 4 can be written as p + q with p, q and p^2 + 4 (or p^2 - 2) all prime. Also, each even number greater than 2 and not equal to 122 can be expressed as p + q with p, q and (p-1)^2 + 1 all prime.

We have verified the first part for n up to 10^8.

Examples

			a(5) = 2 since 5 = 2 + 3 = 3 + 2, and 2*2+1 = 5, 2*3+1 = 7, 2^2+2+1 = 7, 3^2+3+1 = 13 are all prime.
a(31) = 1 since 31 = 14 + 17, and 2*14+1 = 29, 14^2+14+1 = 211 and 17^2+17+1 = 307 are all 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. A002383, A002384, A002375, A219864, A227908, A227909, A230230, A230241, A230254.

Programs

Mathematica

a[n_]:=Sum[If[PrimeQ[2i+1]&&PrimeQ[i^2+i+1]&&PrimeQ[(n-i)^2+n-i+1],1,0],{i,1,n-1}]
Table[a[n],{n,1,100}]