A002384 Numbers m such that m^2 + m + 1 is prime.

1, 2, 3, 5, 6, 8, 12, 14, 15, 17, 20, 21, 24, 27, 33, 38, 41, 50, 54, 57, 59, 62, 66, 69, 71, 75, 77, 78, 80, 89, 90, 99, 101, 105, 110, 111, 117, 119, 131, 138, 141, 143, 147, 150, 153, 155, 161, 162, 164, 167, 168, 173, 176, 188, 189, 192, 194, 203, 206, 209, 215

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Comments

A002383 lists the corresponding primes. - Bernard Schott, Dec 22 2012

This is also the list of bases where 111 represents a prime number. - Christian N. K. Anderson, Mar 28 2013

If d>1 divides m^2 + m + 1, then m + k*d is not in the sequence, for all k>=1. - Gionata Neri, Mar 04 2017

References

• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
• D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
• L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 per l'intervallo compreso entro 121 milioni, e cioè per tutti i valori di n fino a 11000, Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6, v. 3 (1929), pages 193-218.
• 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

• Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
• A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
• Pantelis A. Damianou, On prime values of cyclotomic polynomials, arXiv:1101.1152 [math.NT], Jan 06 2011.
• Oliver Knill, Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes, arXiv preprint arXiv:1606.05958 [math.NT], 2016.
• Oliver Knill, Some experiments in number theory, arXiv preprint arXiv:1606.05971 [math.NT], 2016.

Crossrefs

Cf. A002383, A049407, A049408, A075723, A088503, A110284.

Programs

Magma

[ n: n in [1..300] | IsPrime(n^2+n+1)] // Vincenzo Librandi, Nov 21 2010

Mathematica

Select[Range@ 216, PrimeQ[#^2 + # + 1] &] (* Michael De Vlieger, Mar 06 2017 *)

PARI

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jan 21 2014

Formula

a(n) = (A088503(n) - 1)/2. - Ray Chandler

Extensions

Extended by Ray Chandler, Sep 07 2005