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.

2, 5, 7, 13, 17, 19, 23, 29, 31, 37, 53, 59, 61, 67, 73, 79, 101, 103, 107, 113, 127, 131, 139, 149, 173, 179, 181, 191, 193, 199, 251, 263, 269, 271, 277, 307, 317, 331, 367, 373, 379, 383, 389, 397, 431, 439, 479, 503, 509, 521, 523, 547, 557, 563, 569, 571

Offset: 1

Author

XU Pingya, Aug 31 2020

Keywords

• nonn

Comments

For z <= 10^6, no other prime have this form in the first 105 primes.

Examples

			0^3 - 1^3 + 3*2^3 = 23, 23 is a term.
-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.
		

Crossrefs

Cf. A007490, A221794.

Programs

Mathematica

p1 = Select[Prime[Range[105]], IntegerQ[(# - 1)/3] &];
p2 = Select[Prime[Range[105]], IntegerQ[(# + 1)/3] &];
n1 = Length@p1; n2 = Length@p2;
r1 = (p1 - 1)/3; r2 = (p2 + 1)/3;
t = {};
Do[x = (z^3 + r1[[n]] + 1/4)^(1/2) - 1/2;
 If[IntegerQ[x], AppendTo[t, -x^3 + (x + 1)^3 - 3z^3]], {n, 1,
  n1}, {z, 0, 270}]
Do[x = (z^3 - r2[[n]] + 1/4)^(1/2) - 1/2;
 If[IntegerQ[x], AppendTo[t, x^3 - (x + 1)^3 + 3z^3]], {n, 1,
  n2}, {z, 0, 170}]
Union@t