A219838 Number of ways to write n as x + y with 0 < x <= y and (xy)^2 + xy + 1 prime.

0, 1, 1, 1, 1, 2, 2, 2, 3, 2, 1, 2, 1, 2, 3, 1, 1, 3, 2, 2, 6, 4, 1, 3, 3, 4, 4, 4, 4, 4, 2, 2, 5, 4, 2, 4, 2, 4, 3, 5, 5, 8, 1, 2, 6, 2, 4, 13, 1, 8, 8, 3, 3, 9, 5, 4, 8, 5, 3, 9, 5, 4, 17, 9, 2, 6, 5, 5, 9, 10, 7, 13, 5, 3, 6, 12, 8, 10, 6, 5, 8, 10, 11, 12, 9, 10, 8, 6, 6, 11, 7, 11, 5, 5, 4, 15, 14, 12, 14, 9

Offset: 1

Author

Zhi-Wei Sun, Nov 29 2012

Keywords

• nonn

Comments

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

This has been verified for n up to 10^8. It implies that there are infinitely many primes of the form x^2 + x + 1.

The author also guesses that any integer n > 1157 can be written as x + y with x and y positive integers, and (x*y)^2 + x*y + 1 and (x*y)^2 + x*y - 1 twin primes.

Zhi-Wei Sun has made the following general conjecture: For each prime p, any sufficiently large integer n can be written as x + y, where x and y are positive integers with ((x*y)^p - 1)/(x*y - 1) prime. (For p = 5, 7 it suffices to require n > 28 and n > 46 respectively.)

Compare this with Sun's another conjecture related to A219791.

Examples

			a(49) = 1 since 49 = 3 + 46 with (3*46)^2 + 3*46 + 1 = 19183 prime.
		

Links

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

Crossrefs

Cf. A219791, A091182, A219782.

Programs

Mathematica

a[n_] := a[n] = Sum[If[PrimeQ[k(n - k)(k(n - k) + 1) + 1] == True, 1, 0], {k, 1, n/2}]; Do[Print[n, " ", a[n]], {n, 1, 10000}]