A002328 -id:A002328 - cp's OEIS frontend

A002327 Primes of the form k^2 - k - 1.

5, 11, 19, 29, 41, 71, 89, 109, 131, 181, 239, 271, 379, 419, 461, 599, 701, 811, 929, 991, 1259, 1481, 1559, 1721, 1979, 2069, 2161, 2351, 2549, 2861, 2969, 3079, 3191, 3539, 3659, 4159, 4289, 4421, 4691, 4969, 5851, 6971, 7309, 7481, 8009, 8741, 8929

Offset: 1

N. J. A. Sloane

Also primes of form x*y + x + y or x*y - x - y, where x and y are two successive numbers. - Giovanni Teofilatto, May 12 2004
Equivalently primes p such that 4p+5 is square. - Giovanni Teofilatto, Sep 03 2005
Arithmetic numbers which are triangular, A003601(p)=A000217(k), p prime. sigma_1(p)/sigma_0(p) = k*(k+1)/2; sigma_r(p) divisor function, p prime, k integer. - Ctibor O. Zizka, Jul 14 2008
Also primes of the form k^2 + 3k + 1 (primes in A028387). - Zak Seidov, Apr 13 2014
Also primes p such that the sum of divisors (A000203) of p is oblong (A002378). - Michel Marcus, Jan 09 2015

D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, p. 46.
L. Poletti, Tavole di Numeri Primi Entro Limiti Diversi e Tavole Affini, Milan, 1920, p. 249.
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).

Vincenzo Librandi and Pierre CAMI, Table of n, a(n) for n = 1..10000 (Vincenzo Librandi to n=1000)
Marie Euler and Christophe Petit, Expanding the use of quasi-subfield polynomials, arXiv:1909.11326 [cs.CR], 2019.

Cf. A002328, A088502, A110013, A003601, A000217, A028387.

Cf. A010051.

a002327 n = a002327_list !! (n-1)
a002327_list = filter ((== 1) .  a010051') a028387_list
-- Reinhard Zumkeller, Jul 17 2014

[ a: n in [0..150] | IsPrime(a) where a is n^2 - n - 1 ]; // Vincenzo Librandi, Aug 01 2011

A002327:=n->`if`(isprime(n^2-n-1), n^2-n-1, NULL): seq(A002327(n), n=1..100); # Wesley Ivan Hurt, Aug 09 2014

Select[Table[n^2-n-1,{n,100}],PrimeQ] (* Harvey P. Dale, May 03 2011 *)

for(n=2,1e3,if(isprime(k=n^2-n-1),print1(k", "))) \\ Charles R Greathouse IV, Jul 31 2011

list(lim)=my(v=List(),p); forstep(n=5,sqrtint(4*lim+5),2, if(isprime(p=(n^2-5)/4), listput(v,p))); Vec(v) \\ Charles R Greathouse IV, Oct 10 2023

a(n) = A002328(n)^2 - A002328(n) - 1 = (A110013(n) - 5)/4. - Ray Chandler, Sep 07 2005

a(n) >> n^2 log n by Brun's sieve. - Charles R Greathouse IV, Oct 10 2023

Extended by Ray Chandler, Sep 07 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

A126424 Numbers k for which k^4-k-1 is prime.

Original entry on oeis.org

2, 4, 5, 6, 7, 9, 11, 13, 16, 20, 23, 26, 39, 40, 42, 44, 50, 53, 55, 57, 60, 61, 68, 71, 77, 79, 82, 92, 110, 111, 112, 123, 137, 139, 140, 147, 153, 154, 156, 169, 172, 174, 177, 183, 187, 189, 193, 198, 207, 214, 229, 230, 231, 239, 251, 258, 259, 272, 274, 279

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423.

a = {}; Do[If[PrimeQ[x^4 - x - 1], AppendTo[a, x]], {x, 1, 1000}]; a

A126422 Primes of the form k^4-k-1.

Original entry on oeis.org

13, 251, 619, 1289, 2393, 6551, 14629, 28547, 65519, 159979, 279817, 456949, 2313401, 2559959, 3111653, 3748051, 6249949, 7890427, 9150569, 10555943, 12959939, 13845779, 21381307, 25411609, 35152963, 38950001, 45212093

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002327, A002328, A116581, A126420, A126421, A126423, A126424.

a = {}; Do[If[PrimeQ[x^4 - x - 1], AppendTo[a, x^4 - x - 1]], {x, 1, 100}]; a
Select[Table[n^4-n-1,{n,100}],PrimeQ] (* Harvey P. Dale, Aug 27 2013 *)

a(n) = A126423(A126424(n)). - Amiram Eldar, Mar 13 2020

Definition corrected by Charles R Greathouse IV, Mar 11 2008

A126423 a(n) = n^4 - n - 1.

Original entry on oeis.org

-1, 13, 77, 251, 619, 1289, 2393, 4087, 6551, 9989, 14629, 20723, 28547, 38401, 50609, 65519, 83503, 104957, 130301, 159979, 194459, 234233, 279817, 331751, 390599, 456949, 531413, 614627, 707251, 809969, 923489, 1048543, 1185887, 1336301, 1500589, 1679579, 1874123, 2085097, 2313401, 2559959

Offset: 1

Artur Jasinski, Dec 26 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A002327, A002328, A116581, A126420, A126421, A126422.

[n^4-n-1: n in [1..40]]; // Vincenzo Librandi, Aug 30 2011

a = {}; Do[AppendTo[a, x^4 - x - 1], {x, 1, 100}]; a

From Elmo R. Oliveira, Aug 29 2025: (Start)
G.f.: x*(-1 + 18*x + 2*x^2 + 6*x^3 - x^4)/(1-x)^5.
E.g.f.: 1 + (-1 + 7*x^2 + 6*x^3 + x^4)*exp(x).
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n > 5. (End)

A126426 a(n) = n^5 - n - 1.

Original entry on oeis.org

-1, 29, 239, 1019, 3119, 7769, 16799, 32759, 59039, 99989, 161039, 248819, 371279, 537809, 759359, 1048559, 1419839, 1889549, 2476079, 3199979, 4084079, 5153609, 6436319, 7962599, 9765599, 11881349, 14348879, 17210339, 20511119, 24299969, 28629119, 33554399, 39135359, 45435389

Offset: 1

Artur Jasinski, Dec 26 2006

Every number gives remainder 29 when divided by 30, remainder 9 when divided by 10, and remainder 4 when divided by 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425.

[n^5-n-1: n in [1..40]]; // Vincenzo Librandi, Aug 30 2011

A126426:=n->n^5-n-1: seq(A126426(n), n=1..50); # Wesley Ivan Hurt, Jan 17 2017

a = {}; Do[AppendTo[a, x^5 - x - 1], {x, 1, 100}]; a
Table[n^5-n-1,{n,40}] (* or *) LinearRecurrence[{6,-15,20,-15,6,-1},{-1,29,239,1019,3119,7769},40] (* Harvey P. Dale, Jul 02 2018 *)

a(n)=n^5-n-1 \\ Charles R Greathouse IV, Oct 07 2015

G.f.: x*(x^5-5*x^4+40*x^3+50*x^2+35*x-1)/(1-x)^6. - Colin Barker, Oct 07 2012

A126425 Primes of the form k^5-k-1.

Original entry on oeis.org

29, 239, 1019, 3119, 99989, 161039, 759359, 1048559, 1419839, 2476079, 3199979, 4084079, 14348879, 17210339, 24299969, 45435389, 60466139, 164916179, 254803919, 312499949, 550731719, 1934917559, 2373046799, 3707398349

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

Every number give rest 29 when divided 30, rest 9 when divided 10, rest 4 when divided 5

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126426, A126427.

a = {}; Do[If[PrimeQ[x^5 - x - 1], AppendTo[a, x^5 - x - 1]], {x, 1, 100}]; a
Select[Table[k^5-k-1,{k,90}],PrimeQ] (* Harvey P. Dale, Apr 21 2024 *)

a(n) = A126426(A126427(n)). - Amiram Eldar, Mar 13 2020

A126427 Numbers k for which k^5-k-1 is prime.

Original entry on oeis.org

2, 3, 4, 5, 10, 11, 15, 16, 17, 19, 20, 21, 27, 28, 30, 34, 36, 44, 48, 50, 56, 72, 75, 82, 84, 97, 101, 103, 105, 109, 113, 117, 130, 133, 141, 154, 157, 163, 177, 179, 188, 197, 204, 207, 218, 240, 248, 249, 250, 252, 262, 268, 281, 283, 285, 286, 291, 301, 305, 315

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)

Cf. A002327, A002328, A116581, A126420, A126421, A126422, A126423, A126424, A126425, A126426.

a = {}; Do[If[PrimeQ[x^5 - x - 1], AppendTo[a, x]], {x, 1, 1000}]; a

A094210 Numbers k such that k^2 + 3k + 1 is a prime.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 8, 9, 10, 12, 14, 15, 18, 19, 20, 23, 25, 27, 29, 30, 34, 37, 38, 40, 43, 44, 45, 47, 49, 52, 53, 54, 55, 58, 59, 63, 64, 65, 67, 69, 75, 82, 84, 85, 88, 92, 93, 95, 99, 100, 102, 113, 114, 119, 124, 125, 129, 130, 133, 137, 139, 140, 143, 144, 147, 148

Offset: 1

Giovanni Teofilatto, May 27 2004

Zak Seidov, Table of n, a(n) for n = 1..10000

Cf. A002327, A002328, A028387.

[n: n in [0..200] | IsPrime(n^2 + 3*n + 1)]; // Vincenzo Librandi, Nov 11 2014

Select[ Range[150], PrimeQ[ #^2 + 3# + 1] &] (* Robert G. Wilson v, May 29 2004 *)

a(n) = A002328(n)-2. - R. J. Mathar, Aug 08 2012

Edited and extended by Robert G. Wilson v, May 29 2004

A110013 Squares of the form 4p + 5, where p is a prime.

Original entry on oeis.org

25, 49, 81, 121, 169, 289, 361, 441, 529, 729, 961, 1089, 1521, 1681, 1849, 2401, 2809, 3249, 3721, 3969, 5041, 5929, 6241, 6889, 7921, 8281, 8649, 9409, 10201, 11449, 11881, 12321, 12769, 14161, 14641, 16641, 17161, 17689, 18769, 19881, 23409

Offset: 1

Giovanni Teofilatto, Sep 03 2005

The sequence contain all squares of greater of twin primes.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A002327, A002328, A088502.

Cf. A108604.

Select[4#+5&/@Prime[Range[900]],IntegerQ[Sqrt[#]]&]  (* Harvey P. Dale, Jan 29 2011 *)

a(n) = 4*A002327(n) + 5 = A088502(n)^2.

Corrected and extended by Ray Chandler, Sep 04 2005

cp's OEIS Frontend

A002327 Primes of the form k^2 - k - 1.

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

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A126424 Numbers k for which k^4-k-1 is prime.

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A126422 Primes of the form k^4-k-1.

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A126423 a(n) = n^4 - n - 1.

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A126426 a(n) = n^5 - n - 1.

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A126425 Primes of the form k^5-k-1.

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