A231168 Number of ways to write n = x + y + z (x, y, z > 0) such that x^2 + y^2 + z^2 + z is a square, and 6*x + 1, 6*y - 1, 6*z -1 are all prime.

0, 0, 1, 0, 0, 2, 2, 2, 2, 1, 3, 2, 5, 1, 4, 3, 2, 3, 1, 1, 4, 2, 5, 3, 3, 4, 4, 8, 2, 3, 8, 2, 4, 3, 4, 8, 7, 2, 2, 8, 3, 8, 6, 1, 6, 8, 4, 1, 9, 2, 4, 10, 6, 1, 7, 11, 7, 10, 2, 6, 9, 3, 6, 3, 6, 6, 6, 8, 4, 8, 4, 4, 9, 2, 11, 4, 9, 6, 1, 4, 5, 5, 10, 7, 5, 6, 6, 7, 5, 8, 17, 8, 5, 2, 7, 8, 11, 10, 6, 4

Offset: 1

Author

Zhi-Wei Sun, Nov 04 2013

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n > 5.

Examples

			a(19) = 1 since 19 = 13 + 5 + 1 with 13^2 + 5^2 + 1^2 + 1 = 14^2, and 6*13 + 1 = 79, 6*5 - 1 = 29, 6*1 - 1 = 5 are all prime.
a(444) = 1 since 444 = 76 + 28 + 340 with 76^2 + 28^2 + 340^2 + 340 = 350^2, and 6*76 + 1 = 457, 6*28 - 1 = 167, 6*340 - 1 = 2039 are all prime.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
• Zhi-Wei Sun, A conjecture involving squares and primes, a message to Number Theory List, Nov. 5, 2013.

Crossrefs

Cf. A000040, A000290, A227877, A230121, A230596, A230747.

Programs

Mathematica

SQ[n_]:=IntegerQ[Sqrt[n]]
a[n_]:=Sum[If[PrimeQ[6i+1]&&PrimeQ[6j-1]&&PrimeQ[6(n-i-j)-1]&&SQ[i^2+j^2+(n-i-j)^2+(n-i-j)],1,0],{i,1,n-2},{j,1,n-1-i}]
Table[a[n],{n,1,100}]