A002383 Primes of form k^2 + k + 1. (Formerly M2641 N1051)

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, 22651, 23563

Offset

1,1

Comments

Also primes p such that 4p-3 is square. - Giovanni Teofilatto, 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 Elemer, Apr 15 2003

Also primes of form n^2 - n + 1 (Prime central polygonal numbers, A002061). - Zak Seidov, Jan 26 2006

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

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

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

Conjecture: the set of these numbers, except 3, is the intersection of sets A085104 and A059055. See A225148. - Thomas Ordowski, May 02 2013

For a(n)>13, the fractional part of square root of a(n) starts with digit 5 (see A034101). - Charles Kusniec, Sep 06 2022

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.

H. E. Rose, A Course in Number Theory, Clarendon Press, 1988, p. 217.

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..10751

Cody S. Hansen and Pace P. Nielsen, Prime factors of phi3(x) of the same form, arXiv:2204.08971 [math.NT], 2022.

Bernard Schott, Les nombres brésiliens, Quadrature, no. 76, avril-juin 2010, pages 30-38; included here with permission from the editors of Quadrature.

Formula

a(n) = A002384(n)^2 + A002384(n) + 1 = (A088503(n-1)^2 + 3)/4 = (A110284(n) + 3)/4. - Ray Chandler, Sep 07 2005

Maple

select(isprime, [j^2+j+1$j=1..200])[];  # Alois P. Heinz, Apr 20 2022

Mathematica

Select[Table[n^2+n+1, {n, 250}], PrimeQ] (* Harvey P. Dale, Mar 23 2012 *)

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
(Magma) [ a: n in [1..100] | IsPrime(a) where a is n^2+n+1 ]; // Wesley Ivan Hurt, Jun 16 2014
(Python)
from sympy import isprime
print(list(filter(isprime, (n**2 + n + 1 for n in range(150))))) # Michael S. Branicky, Apr 20 2022

Crossrefs

Cf. A002384, A088503, A110284, A085104.

Cf. A237037, A237038, A237039, A237040 (from semiprimes of form n^3 + 1).

See also A034101.

Sequence in context: A162869 A079018 A342150 * A163418 A309738 A161218

Adjacent sequences: A002380 A002381 A002382 * A002384 A002385 A002386

Keyword

nonn,easy

Author

N. J. A. Sloane

Extensions

Extended by Ray Chandler, Sep 07 2005

Status

approved