A002648 A variant of the cuban primes: primes p = (x^3 - y^3)/(x - y) where x = y + 2. (Formerly M4910 N2105)

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

1,1

Comments

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

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.

Formula

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

Example

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

Mathematica

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

Prog

(PARI) {a(n)= local(m, c); if(n<1, 0, c=0; m=1; while( c<n, m++; if( isprime(m)&issquare((m-1)/3), c++)); m)} /* Michael Somos, Sep 15 2005 */
(Magma) [a: n in [0..400] | IsPrime(a) where a is 3*n^2+1]; // Vincenzo Librandi, Dec 02 2011

Crossrefs

Cf. A002407, A111051 (values of m).

A subsequence of A007645.

Sequence in context: A006239 A271560 A142040 * A055840 A243417 A367592

Adjacent sequences: A002645 A002646 A002647 * A002649 A002650 A002651

Keyword

nonn

Author

N. J. A. Sloane

Extensions

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

Status

approved