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A002383
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Primes of form n^2 + n + 1.
(Formerly M2641 N1051)
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28
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3, 7, 13, 31, 43, 73, 157, 211, 241, 307, 421, 463, 601, 757, 1123, 1483, 1723, 2551, 2971, 3307, 3541, 3907, 4423, 4831, 5113, 5701, 6007, 6163, 6481, 8011, 8191, 9901, 10303, 11131, 12211, 12433, 13807, 14281, 17293, 19183, 20023, 20593, 21757
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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COMMENTS
| Also primes p such that 4p-3 is square. - Giovanni Teofilatto (g.teofilatto(AT)tiscalinet.it), Sep 07 2005
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 E. (labos(AT)ana.sote.hu), Apr 15 2003
Also primes of form n^2 - n + 1 (Prime central polygonal numbers, A002061). - Zak Seidov (zakseidov(AT)yahoo.com), Jan 26 2006
also: Primes of the form : sum of Triangular numbers as: TriangularNumber(n)+TriangularNumber(n+2). 7=1+6, 13=3+10, 31=10+21, 43=15+28, 73=28+45, ... [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 03 2009]
It is not known whether there are infinitely many primes of the form n^2+n+1. H.E.Rose, A Course in Number Theory, Clarendon Press,1988, p. 217. [From Daniel Tisdale (daniel6874(AT)gmail.com), Jun 27 2009]
The finite simple continued fraction [1;n-1,n,n+1] represents the rational number (n^3+2*n+n^2+1)/(n*(n^2+1)); n^2+n+1 is the difference between the numerator and the denominator. [J. M. Bergot, Sep 29 2011]
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REFERENCES
| 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).
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LINKS
| Zak Seidov, Table of n, a(n) for n=1..10751
Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38.
Bernard Schott, Les nombres brésiliens, Reprinted from Quadrature, no. 76, avril-juin 2010, pages 30-38, included here with permission from the editors of Quadrature.
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FORMULA
| a(n) = A002384(n)^2 + A002384(n) + 1 = (A088503(n-1)^2 + 3)/4 = (A110284(n) + 3)/4. - Chandler
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MATHEMATICA
| s=1; Do[s=s+2*n; If[PrimeQ[s], Print[{s, 2*n}]], {n, 1, 100}]
tr[a_]:=Module[{x}, s=0; For[i=1, i<a, s+=i; i++ ]; x=s]; lst={}; Do[a=tr[n]; b=tr[n+2]; p=a+b; If[PrimeQ[p], AppendTo[lst, p]], {n, 4!}]; lst [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 03 2009]
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PROG
| (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
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CROSSREFS
| Cf. A002384, A088503, A110284.
Sequence in context: A083520 A162869 A079018 * A163418 A161218 A068679
Adjacent sequences: A002380 A002381 A002382 * A002384 A002385 A002386
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KEYWORD
| nonn,easy,changed
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| Extended by Ray Chandler (rayjchandler(AT)sbcglobal.net), Sep 07 2005
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