cp's OEIS Frontend

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

%I A219838 #17 Nov 29 2012 14:02:37
%S A219838 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,
%T A219838 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,
%U A219838 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
%N A219838 Number of ways to write n as x + y with 0 < x <= y and (xy)^2 + xy + 1 prime.
%C A219838 Conjecture: a(n) > 0 for all n > 1.
%C A219838 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.
%C A219838 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.
%C A219838 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.)
%C A219838 Compare this with Sun's another conjecture related to A219791.
%H A219838 Zhi-Wei Sun, <a href="/A219838/b219838.txt">Table of n, a(n) for n = 1..10000</a>
%H A219838 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A219838 a(49) = 1 since 49 = 3 + 46 with (3*46)^2 + 3*46 + 1 = 19183 prime.
%t A219838 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}]
%Y A219838 Cf. A219791, A091182, A219782.
%K A219838 nonn
%O A219838 1,6
%A A219838 _Zhi-Wei Sun_, Nov 29 2012