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Indices of prime numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.
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%I #9 Mar 16 2016 16:55:58

%S 1,3,7,12,13,24,25,26,33,35,45,50,51,53,55,59,60,77,79,80,82,84,87,88,

%T 104,106,113,116,121,123,127,135,136,140,148,152,159,165,169,174,176,

%U 178,184,186,189,204,209,211,212,216,218,221,223,226,227,237

%N Indices of prime numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

%C How is the distribution of a(n), a(n+1) in this sequence where a(n+1) = a(n) + 1?

%e 1 is a term because prime(1) = 2 = (-2)^2 + (-2) + (-2)*1 + 1 + 1^2.

%e 3 is a term because prime(3) = 5 = 1^2 + 1 + 1*1 + 1 + 1^2.

%e 7 is a term because prime(7) = 17 = (-5)^2 + (-5) + (-5)*3 + 3 + 3^2.

%e 12 is a term because prime(12) = 37 = (-7)^2 + (-7) + (-7)*5 + 5 + 5^2.

%e 13 is a term because prime(13) = 41 = 7^2 + 7 + 7*(-3) + (-3) + (-3)^2.

%o (PARI) isA003136(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);

%o for(n=1, 250, if(isA003136(3*prime(n)+1), print1(n, ", ")));

%Y Cf. A000040, A003136, A267137.

%K nonn

%O 1,2

%A _Altug Alkan_, Mar 11 2016