

A002384


Numbers n such that n^2 + n + 1 is prime.
(Formerly M0626 N0228)


68



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Damianou, p. 4, claims that there is strong experimental evidence that n^2+n+1 is prime for infinitely many values of n. There are a total of 189 such values under 1000.  Jonathan Vos Post, Jan 06 2011
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 n^2 + n + 1, then n + k*d is not in the sequence, for all k>=1.  Gionata Neri, Mar 04 2017


REFERENCES

A. J. C. Cunningham, Binomial Factorisations, Vols. 19, Hodgson, London, 19231929; see Vol. 1, pp. 245259.
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+n1 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 193218.
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

T. D. Noe and 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. 19, Hodgson, London, 19231929. [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.


FORMULA

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


MATHEMATICA

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


PROG

(MAGMA)[ n: n in [1..300]  IsPrime(n^2+n+1)] // Vincenzo Librandi, Nov 21 2010
(PARI) is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jan 21 2014


CROSSREFS

Cf. A002383, A049407, A049408, A075723, A088503, A110284.
Sequence in context: A120768 A271109 A293033 * A096176 A002243 A094763
Adjacent sequences: A002381 A002382 A002383 * A002385 A002386 A002387


KEYWORD

nonn,easy


AUTHOR

N. J. A. Sloane


EXTENSIONS

Extended by Ray Chandler, Sep 07 2005


STATUS

approved



