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%I M2641 N1051 #109 Sep 07 2022 18:58:14
%S 3,7,13,31,43,73,157,211,241,307,421,463,601,757,1123,1483,1723,2551,
%T 2971,3307,3541,3907,4423,4831,5113,5701,6007,6163,6481,8011,8191,
%U 9901,10303,11131,12211,12433,13807,14281,17293,19183,20023,20593,21757,22651,23563
%N Primes of form k^2 + k + 1.
%C Also primes p such that 4p-3 is square. - _Giovanni Teofilatto_, Sep 07 2005
%C Also these primes are sums of 1 and some consecutive even numbers starting at 2; e.g., 31 = 1+2+4+6+8+10. - _Labos Elemer_, Apr 15 2003
%C Also primes of form n^2 - n + 1 (Prime central polygonal numbers, A002061). - _Zak Seidov_, Jan 26 2006
%C Also primes which are of the form TriangularNumber(n) + TriangularNumber(n+2): 7 = 1+6, 13 = 3+10, 31 = 10+21, 43 = 15+28, 73 = 28+45, ... - _Vladimir Joseph Stephan Orlovsky_, Apr 03 2009
%C It is not known whether there are infinitely many primes of the form n^2+n+1. See Rose reference. - _Daniel Tisdale_, Jun 27 2009
%C These numbers when >= 7 are prime repunits 111_n in a base n >= 2, so except for 3, they are all Brazilian primes belonging to A085104. (See Links "Les nombres brésiliens", Sections V.4 - V.5.) A002383 is generated by A002384 which lists the bases n of 111_n. A002383 = A053183 Union A185632. - _Bernard Schott_, Dec 22 2012
%C Conjecture: the set of these numbers, except 3, is the intersection of sets A085104 and A059055. See A225148. - _Thomas Ordowski_, May 02 2013
%C For a(n)>13, the fractional part of square root of a(n) starts with digit 5 (see A034101). - _Charles Kusniec_, Sep 06 2022
%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 H. E. Rose, A Course in Number Theory, Clarendon Press, 1988, p. 217.
%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="/A002383/b002383.txt">Table of n, a(n) for n = 1..10751</a>
%H Cody S. Hansen and Pace P. Nielsen, <a href="https://arxiv.org/abs/2204.08971">Prime factors of phi3(x) of the same form</a>, arXiv:2204.08971 [math.NT], 2022.
%H Bernard Schott, <a href="/A125134/a125134.pdf">Les nombres brésiliens</a>, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.
%F a(n) = A002384(n)^2 + A002384(n) + 1 = (A088503(n-1)^2 + 3)/4 = (A110284(n) + 3)/4. - _Ray Chandler_, Sep 07 2005
%p select(isprime, [j^2+j+1$j=1..200])[]; # _Alois P. Heinz_, Apr 20 2022
%t Select[Table[n^2+n+1, {n,250}], PrimeQ] (* _Harvey P. Dale_, Mar 23 2012 *)
%o (PARI) list(lim)=select(n->isprime(n),vector((sqrt(4*lim-3)-1)\2,k,k^2+k+1)) \\ _Charles R Greathouse IV_, Jul 25 2011
%o (Magma) [ a: n in [1..100] | IsPrime(a) where a is n^2+n+1 ]; // _Wesley Ivan Hurt_, Jun 16 2014
%o (Python)
%o from sympy import isprime
%o print(list(filter(isprime, (n**2 + n + 1 for n in range(150))))) # _Michael S. Branicky_, Apr 20 2022
%Y Cf. A002384, A088503, A110284, A085104.
%Y Cf. A237037, A237038, A237039, A237040 (from semiprimes of form n^3 + 1).
%Y See also A034101.
%K nonn,easy
%O 1,1
%A _N. J. A. Sloane_
%E Extended by _Ray Chandler_, Sep 07 2005
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