%I M0626 N0228 #78 Oct 28 2023 11:47:51
%S 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,
%T 75,77,78,80,89,90,99,101,105,110,111,117,119,131,138,141,143,147,150,
%U 153,155,161,162,164,167,168,173,176,188,189,192,194,203,206,209,215
%N Numbers m such that m^2 + m + 1 is prime.
%C A002383 lists the corresponding primes. - _Bernard Schott_, Dec 22 2012
%C This is also the list of bases where 111 represents a prime number. - _Christian N. K. Anderson_, Mar 28 2013
%C 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
%D A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
%D D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
%D 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.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Zak Seidov, <a href="/A002384/b002384.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from T. D. Noe)
%H A. J. C. Cunningham, <a href="/A001912/a001912.pdf">Binomial Factorisations</a>, Vols. 1-9, Hodgson, London, 1923-1929. [Annotated scans of a few pages from Volumes 1 and 2]
%H Pantelis A. Damianou, <a href="http://arxiv.org/abs/1101.1152">On prime values of cyclotomic polynomials</a>, arXiv:1101.1152 [math.NT], Jan 06 2011.
%H Oliver Knill, <a href="https://arxiv.org/abs/1606.05958">Goldbach for Gaussian, Hurwitz, Octavian and Eisenstein primes</a>, arXiv preprint arXiv:1606.05958 [math.NT], 2016.
%H Oliver Knill, <a href="https://arxiv.org/abs/1606.05971">Some experiments in number theory</a>, arXiv preprint arXiv:1606.05971 [math.NT], 2016.
%F a(n) = (A088503(n) - 1)/2. - _Ray Chandler_
%t Select[Range@ 216, PrimeQ[#^2 + # + 1] &] (* _Michael De Vlieger_, Mar 06 2017 *)
%o (Magma)[ n: n in [1..300] | IsPrime(n^2+n+1)] // _Vincenzo Librandi_, Nov 21 2010
%o (PARI) is(n)=isprime(n^2+n+1) \\ _Charles R Greathouse IV_, Jan 21 2014
%Y Cf. A002383, A049407, A049408, A075723, A088503, A110284.
%K nonn,easy
%O 1,2
%A _N. J. A. Sloane_
%E Extended by _Ray Chandler_, Sep 07 2005