A002640 Numbers k such that (k^2 + k + 1)/3 is prime.

4, 7, 10, 13, 19, 28, 31, 34, 40, 43, 52, 70, 73, 76, 82, 85, 91, 97, 103, 112, 115, 124, 127, 136, 145, 148, 157, 166, 175, 187, 190, 199, 202, 223, 241, 244, 259, 265, 271, 274, 280, 286, 316, 325, 358, 370, 376, 385, 388, 409, 421, 427, 442, 460, 469, 472

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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..1000
• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
• Hajrudin Fejzić, Dan Rinne, and Bob Stein, On Sums of Cubes, The College Mathematics Journal, Vol. 36, No. 3 (May, 2005), pp. 226-228. See p. 228.

Crossrefs

Cf. A002384.

Programs

Magma

[n: n in [4..500] | IsPrime((n^2+n+1) div 3)]; // Vincenzo Librandi, Nov 18 2010

Mathematica

Select[Range[500], PrimeQ[(#^2 + # + 1)/3] &] (* Vincenzo Librandi, Sep 25 2012 *)

PARI

isok(k) = my(x=k^2+k+1); !(x%3) && isprime(x/3); \\ Michel Marcus, Aug 22 2025