A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2.

13, 109, 193, 433, 769, 1201, 1453, 2029, 3469, 3889, 4801, 10093, 12289, 13873, 18253, 20173, 21169, 22189, 28813, 37633, 43201, 47629, 60493, 63949, 65713, 69313, 73009, 76801, 84673, 106033, 108301, 112909, 115249, 129793, 139969, 142573, 147853, 169933

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Primes p such that p = 1 + 3*m^2 for some integer m (A111051). - Michael Somos, Sep 15 2005

Examples

			193 is a term since 193 = (9^3 - 7^3)/(9 - 7) is a prime.
		

References

• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..5000
• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
• Eric Weisstein's World of Mathematics, Cuban Prime.
• Wikipedia, Cuban prime.

Crossrefs

Cf. A002407, A111051 (values of m).

A subsequence of A007645.

Programs

Magma

[a: n in [0..400] | IsPrime(a) where a is 3*n^2+1]; // Vincenzo Librandi, Dec 02 2011

Mathematica

Select[Table[3n^2+1,{n,0,700}],PrimeQ] (* Vincenzo Librandi, Dec 02 2011 *)

PARI

{a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( cMichael Somos, Sep 15 2005 */

Formula

a(n) = 3*A111051(n)^2 + 1. - Paul F. Marrero Romero, Nov 03 2023

Extensions

Entry revised by N. J. A. Sloane, Jan 29 2013