A002384 -id:A002384 - cp's OEIS frontend

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

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

A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

Original entry on oeis.org

2, 3, 5, 10, 11, 12, 16, 20, 21, 22, 33, 37, 38, 43, 47, 48, 55, 71, 75, 76, 80, 81, 111, 121, 126, 131, 133, 135, 136, 141, 155, 157, 158, 165, 176, 177, 180, 203, 223, 242, 245, 251, 253, 256, 257, 258, 265, 268, 276, 286, 290, 297, 307, 322, 323, 342, 361, 363, 366, 375, 377, 385, 388, 396, 411

Offset: 1

Eric Chen, Nov 13 2014

nonn

Numbers n such that (n^5+1)/(n+1) is prime, or numbers n such that A060884(n) is prime.

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 893 terms from Robert Price)
OEIS Wiki, Cyclotomic Polynomials at x=sigma(n)

Cf. A008864 (1), A006093 (2), A002384 (3), A005574 (4), A049409 (5), A055494 (6), A100330 (7), A000068 (8), A153439 (9), this sequence (10), A162862 (11), A246397 (12), A217070 (13), A006314 (16), A217071 (17), A164989 (18), A217072 (19), A217073 (23), A153440 (27), A217074 (29), A217075 (31), A006313 (32), A097475 (36), A217076 (37), A217077 (41), A217078 (43), A217079 (47), A217080 (53), A217081 (59), A217082 (61), A006315 (64), A217083 (67), A217084 (71), A217085 (73), A217086 (79), A153441 (81), A217087 (83), A217088 (89), A217089 (97), A006316 (128), A153442 (243), A056994 (256), A056995 (512), A057465 (1024), A057002 (2048), A088361 (4096), A088362 (8192), A226528 (16384), A226529 (32768), A226530 (65536).

Cf. A060884, A085398, A259257.

[n: n in [1..500]| IsPrime((n^5+1) div (n+1))]; // Vincenzo Librandi, Nov 14 2014

A246392:=n->`if`(isprime((n^5+1)/(n+1)),n,NULL): seq(A246392(n), n=1..500); # Wesley Ivan Hurt, Nov 15 2014

Select[Range[700], PrimeQ[(#^5 + 1) / (# + 1)] &] (* Vincenzo Librandi, Nov 14 2014 *)

for(n=1,10^3,if(isprime(polcyclo(10,n)),print1(n,", "))); \\ Joerg Arndt, Nov 13 2014

A049407 Numbers m such that m^3 + m + 1 is prime.

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 12, 15, 17, 18, 21, 29, 30, 32, 39, 41, 42, 44, 48, 53, 54, 56, 60, 69, 71, 74, 77, 83, 87, 95, 102, 104, 108, 116, 117, 120, 126, 131, 135, 143, 144, 146, 152, 153, 155, 162, 168, 177, 179, 180, 186, 191, 207, 212, 219, 221, 225, 239, 240, 243

Offset: 1

N. J. A. Sloane

For s = 5, 8, 11, 14, 17, 20, ... (A016789(s) for s>=2), m_s = 1 + m + m^s is composite for m>1. Also for m=1, m_s = 3 is a prime for any s. Here we consider the case s=3.
If m == 1 (mod 3), m_s == 0 (mod 3) for any s and is not prime for m > 1. Thus for n > 1, a(n) !== 1 (mod 3) and this is true for any similar sequence based on another s value (A002384, A049408, A075723). - Jean-Christophe Hervé, Sep 20 2014
Corresponding primes are in A095692.

			3 is a term because 1 + 3 + 3^3 = 31 is a prime.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002384 (s=2), A049408 (s=4), A075723 (s=6).

Cf. A095692 (corresponding primes).

[n: n in [0..300] | IsPrime(s) where s is 1+&+[n^i: i in [1..3 by 2]]]; // Vincenzo Librandi, Jun 27 2014

A049407:=n->`if`(isprime(n^3+n+1), n, NULL): seq(A049407(n), n=1..300); # Wesley Ivan Hurt, Nov 14 2014

Select[Range[500], PrimeQ[Total[#^Range[1, 3, 2]] + 1] &] (* Vincenzo Librandi, Jun 27 2014 *)

is(n)=isprime(n^3+n+1) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import isprime
def ok(m): return isprime(m**3 + m + 1)
print([m for m in range(244) if ok(m)]) # Michael S. Branicky, Feb 17 2022

A055494 Numbers k such that k^2 - k + 1 is prime.

Original entry on oeis.org

2, 3, 4, 6, 7, 9, 13, 15, 16, 18, 21, 22, 25, 28, 34, 39, 42, 51, 55, 58, 60, 63, 67, 70, 72, 76, 78, 79, 81, 90, 91, 100, 102, 106, 111, 112, 118, 120, 132, 139, 142, 144, 148, 151, 154, 156, 162, 163, 165, 168, 169, 174, 177, 189, 190, 193, 195, 204, 207, 210, 216

Offset: 1

Robert G. Wilson v, Jul 05 2000

Also nonnegative integers such that a(n) + w is an Eisenstein prime, where w is the primitive cube root of unity. - Frank M Jackson, Jul 15 2025

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988.
D. E. Knuth, "The Art of Computer Programming," Addison-Wesley, Reading, MA, Volume II, page 378.
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Eisenstein Prime.

Equals A002384(n)+1.

A002383 gives the primes.

[n: n in [1..300] |IsPrime(n^2 - n + 1)]; // Vincenzo Librandi, Sep 28 2012

Select[Range[300], PrimeQ[#^2 - # + 1] &] (* Vincenzo Librandi, Sep 28 2012 *)
lst = {}; Do[If[ResourceFunction["EisensteinIntegers"][n + w, w, "PrimeQ"],AppendTo[lst, n]], {n, 1, 300}]; lst (* Frank M Jackson, Jul 15 2025 *)

is(n)=isprime(n^2-n+1) \\ Charles R Greathouse IV, Feb 16 2017

More terms from David Wasserman, Sep 15 2005

A045546 Numbers k such that k^2 + k - 1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 6, 8, 9, 10, 11, 13, 15, 16, 19, 20, 21, 24, 26, 28, 30, 31, 35, 38, 39, 41, 44, 45, 46, 48, 50, 53, 54, 55, 56, 59, 60, 64, 65, 66, 68, 70, 76, 83, 85, 86, 89, 93, 94, 96, 100, 101, 103, 114, 115, 120, 125, 126, 130, 131, 134, 138, 140

Offset: 1

Paul Jobling (paul.jobling(AT)whitecross.com)

L. Poletti, Le serie dei numeri primi appartenente alle due forme quadratiche (A) n^2+n+1 e (B) n^2+n-1 ..., Atti della Reale Accademia Nazionale dei Lincei, Memorie della Classe di Scienze Fisiche, Matematiche e Naturali, s. 6. 3 (1929), 193-218.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Vincenzo Librandi)

Equals A002328-1. Cf. A002327, A002384.

[n: n in [0..140] | IsPrime(n^2 + n - 1)]; // Vincenzo Librandi, Sep 27 2012

lst={};Do[If[PrimeQ[n^2+n-1], Print[n];AppendTo[lst, n]], {n, 10^5}];lst (* Vladimir Joseph Stephan Orlovsky, Aug 25 2008 *)
Select[Range[150],PrimeQ[#^2+#-1]&] (* Harvey P. Dale, Jan 27 2012 *)

is(n)=isprime(n^2 + n - 1) \\ Charles R Greathouse IV, Apr 28 2015

A217070 Numbers k such that (k^13-1)/(k-1) is prime.

Original entry on oeis.org

2, 3, 5, 7, 34, 37, 43, 59, 72, 94, 98, 110, 133, 149, 151, 159, 190, 207, 219, 221, 251, 260, 264, 267, 282, 286, 291, 319, 355, 363, 373, 382, 397, 398, 402, 406, 408, 412, 436, 442, 486, 489, 507, 542, 544, 552, 553, 582, 585, 592, 603, 610, 614, 634, 643

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A002384, A049409, A100330, A162862, A217071-A217089.

[n: n in [2..1000] |IsPrime((n^13 - 1) div (n - 1))]; // Vincenzo Librandi, Sep 28 2012

Select[Range[2, 1000], PrimeQ[(#^13 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime(polcyclo(13,n)) \\ Charles R Greathouse IV, Apr 28 2015

A217089 Numbers n such that (n^97-1)/(n-1) is prime.

Original entry on oeis.org

12, 90, 104, 234, 271, 339, 420, 421, 428, 429, 464, 805, 909, 934, 1054, 1114, 1116, 1128, 1144, 1159, 1193, 1364, 1788, 2086, 2215, 2254, 2448, 2461, 2593, 2595, 2771, 2787, 2829, 2859, 2952, 3029, 3075, 3144, 3250, 3265, 3268, 3301, 3701, 3752, 3875, 4026

Offset: 1

Tim Johannes Ohrtmann, Sep 26 2012

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A002384, A049409, A100330, A162862, A217070-A217088.

Select[Range[2, 1000], PrimeQ[(#^97 - 1)/(# - 1)] &] (* T. D. Noe, Sep 26 2012 *)

is(n)=isprime((n^97-1)/(n-1)) \\ Charles R Greathouse IV, Feb 17 2017

More terms from T. D. Noe, Sep 26 2012

A085398 Let Cn(x) be the n-th cyclotomic polynomial; a(n) is the least k>1 such that Cn(k) is prime.

Original entry on oeis.org

3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 5, 2, 2, 2, 2, 2, 2, 6, 2, 4, 3, 2, 10, 2, 22, 2, 2, 4, 6, 2, 2, 2, 2, 2, 14, 3, 61, 2, 10, 2, 14, 2, 15, 25, 11, 2, 5, 5, 2, 6, 30, 11, 24, 7, 7, 2, 5, 7, 19, 3, 2, 2, 3, 30, 2, 9, 46, 85, 2, 3, 3, 3, 11, 16, 59, 7, 2, 2, 22, 2, 21, 61, 41, 7, 2, 2, 8, 5, 2, 2

Offset: 1

Don Reble, Jun 28 2003

nonn

Conjecture: a(n) is defined for all n. - Eric Chen, Nov 14 2014

Existence of a(n) is implied by Bunyakovsky's conjecture. - Robert Israel, Nov 13 2014

			a(11) = 5 because C11(k) is composite for k = 2, 3, 4 and prime for k = 5.
a(37) = 61 because C37(k) is composite for k = 2, 3, 4, ..., 60 and prime for k = 61.

Jinyuan Wang, Table of n, a(n) for n = 1..5000 (terms 1..1500 from Eric Chen)
Wikipedia, Bunyakowsky conjecture

Cf. A117544, A066180, A085399, A103795, A056993, A153438, A246119, A246120, A246121, A206418, A205506, A181980.

Cf. A008864, A006093, A002384, A005574, A049409, A055494, A100330, A000068, A153439, A246392, A162862, A246397, A217070, A006314, A217071, A164989, A217072, A217073, A153440, A217074, A217075, A006313, A097475.

f:= proc(n) local k;
for k from 2 do if isprime(numtheory:-cyclotomic(n,k)) then return k fi od
end proc:
seq(f(n), n = 1 .. 100); # Robert Israel, Nov 13 2014

Table[k = 2; While[!PrimeQ[Cyclotomic[n, k]], k++]; k, {n, 300}] (* Eric Chen, Nov 14 2014 *)

a(n) = k=2; while(!isprime(polcyclo(n, k)), k++); k; \\ Michel Marcus, Nov 13 2014

a(A072226(n)) = 2. - Eric Chen, Nov 14 2014
a(n) = A117544(n) except when n is a prime power, since if n is a prime power, then A117544(n) = 1. - Eric Chen, Nov 14 2014
a(prime(n)) = A066180(n), a(2*prime(n)) = A103795(n), a(2^n) = A056993(n-1), a(3^n) = A153438(n-1), a(2*3^n) = A246120(n-1), a(3*2^n) = A246119(n-1), a(6^n) = A246121(n-1), a(5^n) = A206418(n-1), a(6*A003586(n)) = A205506(n), a(10*A003592(n)) = A181980(n).

A053183 Primes of the form p^2 + p + 1 when p is prime.

Original entry on oeis.org

7, 13, 31, 307, 1723, 3541, 5113, 8011, 10303, 17293, 28057, 30103, 86143, 147073, 459007, 492103, 552793, 579883, 598303, 684757, 704761, 735307, 830833, 1191373, 1204507, 1353733, 1395943, 1424443, 1482307, 1886503, 2037757

Offset: 1

Enoch Haga, Mar 01 2000

Also primes in A001001. - Philippe Deléham, Feb 21 2004
This sequence is a subsequence of A002383. These numbers are repunit primes 111_n, so they are Brazilian primes belonging to A085104. - Bernard Schott, Dec 21 2012
Also, primes in A060800. - Zak Seidov, Mar 21 2014
Also subsequence of A002061, A193574. - Hartmut F. W. Hoft, May 05 2017
As p^2 + p + 1 is the sum of divisors of p^2 for any prime p, this is a subsequence of A023195. - Peter Munn, Feb 16 2018

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A002383, A002384, A023195, A053182, A060800, A065764, A085104, A182253, A185632.

Cf. A002061, A193574. - Hartmut F. W. Hoft, May 05 2017

a053183[n_] := Select[Map[Prime[#]^2 + Prime[#] + 1&, Range[n]], PrimeQ]
a053183[225] (* data *) (* Hartmut F. W. Hoft, May 05 2017 *)
Select[Table[p^2+p+1,{p,Prime[Range[300]]}],PrimeQ] (* Harvey P. Dale, Aug 15 2017 *)

a(n) = A053182(n)^2 + A053182(n) + 1.

A049408 Numbers k such that k^4 + k + 1 is prime.

Original entry on oeis.org

1, 2, 5, 6, 9, 11, 12, 14, 24, 26, 32, 36, 44, 47, 60, 69, 72, 74, 77, 89, 90, 102, 107, 119, 126, 131, 146, 147, 159, 162, 170, 171, 186, 191, 197, 204, 206, 219, 239, 240, 252, 266, 284, 285, 290, 296, 300, 324, 347, 351, 362, 384, 426, 437, 459, 465, 470

Offset: 1

N. J. A. Sloane

For s = 5,8,11,14,17,20,..., n_s = 1 + n + n^s is always composite for any n > 1. Also for n=1, n_s=3 is a prime for any s. Here we consider the case s=4.

			26 is a term because at s=4, n=26, n_s = 1 + n + n^s = 457003 is a prime.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A002384, A075723, A049407.

[n: n in [0..1000] | IsPrime(s) where s is 1+n+n^4]; // Vincenzo Librandi, Jul 28 2014

Select[Range[1000], PrimeQ[1 + # + #^4] &] (* Vincenzo Librandi, Jul 28 2014 *)

for(n=1, 1000, if(isprime(1+n+n^4), print1(n",")))

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

Formula

Extensions

A246392 Numbers n such that Phi(10, n) is prime, where Phi is the cyclotomic polynomial.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

A049407 Numbers m such that m^3 + m + 1 is prime.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Python

A055494 Numbers k such that k^2 - k + 1 is prime.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

Extensions

A045546 Numbers k such that k^2 + k - 1 is prime.

Views

Author

Keywords

References

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A217070 Numbers k such that (k^13-1)/(k-1) is prime.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Mathematica

PARI

A217089 Numbers n such that (n^97-1)/(n-1) is prime.

Views

Author