A088503 -id:A088503 - cp's OEIS frontend

A002384 Numbers m such that m^2 + m + 1 is prime.

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

Offset: 1

N. J. A. Sloane

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 m^2 + m + 1, then m + k*d is not in the sequence, for all k>=1. - Gionata Neri, Mar 04 2017

A. J. C. Cunningham, Binomial Factorisations, Vols. 1-9, Hodgson, London, 1923-1929; see Vol. 1, pp. 245-259.
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).

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. 1-9, Hodgson, London, 1923-1929. [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.

Cf. A002383, A049407, A049408, A075723, A088503, A110284.

[ n: n in [1..300] | IsPrime(n^2+n+1)] // Vincenzo Librandi, Nov 21 2010

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

is(n)=isprime(n^2+n+1) \\ Charles R Greathouse IV, Jan 21 2014

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

Extended by Ray Chandler, Sep 07 2005

A002383 Primes of form k^2 + k + 1.

Original entry on oeis.org

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

N. J. A. Sloane

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

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).

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.

Cf. A002384, A088503, A110284, A085104.
Cf. A237037, A237038, A237039, A237040 (from semiprimes of form n^3 + 1).
See also A034101.

[ a: n in [1..100] | IsPrime(a) where a is n^2+n+1 ]; // Wesley Ivan Hurt, Jun 16 2014

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

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

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

from sympy import isprime
print(list(filter(isprime, (n**2 + n + 1 for n in range(150))))) # Michael S. Branicky, Apr 20 2022

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

Extended by Ray Chandler, Sep 07 2005

A110284 Squares of the form 4p - 3, where p is a prime.

Original entry on oeis.org

9, 25, 49, 121, 169, 289, 625, 841, 961, 1225, 1681, 1849, 2401, 3025, 4489, 5929, 6889, 10201, 11881, 13225, 14161, 15625, 17689, 19321, 20449, 22801, 24025, 24649, 25921, 32041, 32761, 39601, 41209, 44521, 48841, 49729, 55225, 57121, 69169

Offset: 1

Giovanni Teofilatto, Sep 07 2005

nonn

Also: squares of the form 2*s-3, where s is a semiprime, A107317. - Franklin T. Adams-Watters, Jun 28 2010

Squares are less dense then primes and easy to generate so it's faster to check squares if they are of the required form than to check if primes are of the required form. - David A. Corneth, Oct 15 2018

David A. Corneth, Table of n, a(n) for n = 1..16289 (First 1316 terms by Marius A. Burtea, terms < 10^11)

Cf. A002383, A002384, A088503, A001358.

[4*p - 3: p in PrimesUpTo(10^5)|IsSquare (4*p - 3)]; // Vincenzo Librandi, Oct 17 2018

Select[ 4Prime[ Range[2000]] - 3, IntegerQ[ Sqrt[ # ]] &] (* Robert G. Wilson v, Sep 20 2005 *)

isok(n) = issquare(n) && (p=(n+3)/4) && (frac(p)==0) && isprime(p); \\ Michel Marcus, Oct 15 2018

upto(n) = my(res = List()); forstep(i = 3, sqrtint(n), 2, if(isprime((i^2+3)/4), listput(res, i^2))); res \\ David A. Corneth, Oct 15 2018

a(n) = 4*A002383(n) - 3 = A088503(n-1)^2.

Extended by Ray Chandler, Sep 07 2005

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A002384 Numbers m such that m^2 + m + 1 is prime.

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A002383 Primes of form k^2 + k + 1.

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A110284 Squares of the form 4p - 3, where p is a prime.

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Crossrefs

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Magma

Mathematica

PARI

PARI

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Extensions