cp's OEIS Frontend

A336451 Primes of form x^3 - (x + 1)^3 + 3*z^3 or -x^3 + (x + 1)^3 - 3*z^3, with x,z >= 0.

%I A336451 #4 Sep 20 2020 01:50:41
%S A336451 2,5,7,13,17,19,23,29,31,37,53,59,61,67,73,79,101,103,107,113,127,131,
%T A336451 139,149,173,179,181,191,193,199,251,263,269,271,277,307,317,331,367,
%U A336451 373,379,383,389,397,431,439,479,503,509,521,523,547,557,563,569,571
%N A336451 Primes of form x^3 - (x + 1)^3 + 3*z^3 or -x^3 + (x + 1)^3 - 3*z^3, with x,z >= 0.
%C A336451 For z <= 10^6, no other prime have this form in the first 105 primes.
%e A336451 0^3 - 1^3 + 3*2^3 = 23, 23 is a term.
%e A336451 -3^3 + 4^3 - 3*0^3 = -4^3 + 5^3 - 3*2^3 = -52^3 + 53^3 - 3*14^3 = 37, 37 is a term.
%t A336451 p1 = Select[Prime[Range[105]], IntegerQ[(# - 1)/3] &];
%t A336451 p2 = Select[Prime[Range[105]], IntegerQ[(# + 1)/3] &];
%t A336451 n1 = Length@p1; n2 = Length@p2;
%t A336451 r1 = (p1 - 1)/3; r2 = (p2 + 1)/3;
%t A336451 t = {};
%t A336451 Do[x = (z^3 + r1[[n]] + 1/4)^(1/2) - 1/2;
%t A336451  If[IntegerQ[x], AppendTo[t, -x^3 + (x + 1)^3 - 3z^3]], {n, 1,
%t A336451   n1}, {z, 0, 270}]
%t A336451 Do[x = (z^3 - r2[[n]] + 1/4)^(1/2) - 1/2;
%t A336451  If[IntegerQ[x], AppendTo[t, x^3 - (x + 1)^3 + 3z^3]], {n, 1,
%t A336451   n2}, {z, 0, 170}]
%t A336451 Union@t
%Y A336451 Cf. A007490, A221794.
%K A336451 nonn
%O A336451 1,1
%A A336451 _XU Pingya_, Aug 31 2020