A126420 -id:A126420 - cp's OEIS frontend

A126421 Numbers n for which n^3-n-1 is prime.

2, 3, 4, 8, 9, 11, 14, 15, 16, 18, 21, 23, 24, 29, 31, 35, 36, 38, 41, 45, 48, 55, 58, 59, 60, 64, 69, 70, 71, 76, 81, 86, 98, 104, 108, 111, 113, 115, 119, 123, 126, 128, 130, 136, 140, 150, 151, 155, 163, 168, 174, 176, 183, 184, 185, 186, 188, 189, 191, 203, 206

Offset: 1

Artur Jasinski, Dec 26 2006

nonn

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

Cf. A002327, A116581, A126420.

a = {}; Do[If[PrimeQ[x^3 - x - 1], AppendTo[a, x]], {x, 1, 1000}]; a
Select[Range[300],PrimeQ[#^3-#-1]&] (* Harvey P. Dale, Jul 22 2013 *)

is(n)=isprime(n^3-n-1) \\ Charles R Greathouse IV, Apr 29 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

A116581 Primes of the form k^3 - k - 1.

Original entry on oeis.org

5, 23, 59, 503, 719, 1319, 2729, 3359, 4079, 5813, 9239, 12143, 13799, 24359, 29759, 42839, 46619, 54833, 68879, 91079, 110543, 166319, 195053, 205319, 215939, 262079, 328439, 342929, 357839, 438899, 531359, 635969, 941093, 1124759, 1259603, 1367519, 1442783

Offset: 1

Roger L. Bagula, Mar 22 2006

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..10000
Ken Ono and Scott Ahlgren, Weierstrass points on X0(p) and supersingular j-invariants, Mathematische Annalen 325, 2003, pp. 355-368.

Cf. A002327, A126420, A126421.

[ a: n in [1..200] | IsPrime(a) where a is n^3-n-1 ]; // Vincenzo Librandi, Dec 07 2011

Select[Table[n^3-n-1,{n,0,800}],PrimeQ] (* Vincenzo Librandi, Dec 07 2011 *)

from sympy import isprime
def aupton(terms):
  k, alst = 2, []
  while len(alst) < terms:
    if isprime(k**3-k-1): alst.append(k**3-k-1)
    k += 1
  return alst
print(aupton(37)) # Michael S. Branicky, May 23 2021

a(n) = A126420(A126421(n)). - Elmo R. Oliveira, Apr 20 2025

Edited by N. J. A. Sloane, Jan 01 2007

More terms from Artur Jasinski, Jan 01 2007

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

A061600 a(n) = n^3 - n + 1.

Original entry on oeis.org

1, 1, 7, 25, 61, 121, 211, 337, 505, 721, 991, 1321, 1717, 2185, 2731, 3361, 4081, 4897, 5815, 6841, 7981, 9241, 10627, 12145, 13801, 15601, 17551, 19657, 21925, 24361, 26971, 29761, 32737, 35905, 39271, 42841, 46621, 50617, 54835, 59281, 63961

Offset: 0

Amarnath Murthy, May 19 2001

Smallest of n consecutive odd numbers whose sum is n^4. (n^k can be expressed as the sum of n consecutive odd numbers the smallest of which is given by n^(k-1)-n+1.)

			a(5) = 121 = 5^3 - 5 + 1. We have 121 + 123 + 125 + 127 + 129 = 625 = 5^4.

T. A. Gulliver, Sequences from Cubes of Integers, Int. Math. Journal, 4 (2003), 439-445.

Harry J. Smith, Table of n, a(n) for n = 0..1000
Leo Tavares, Illustration: Diamond Chains
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A126420.

[n^3 - n + 1: n in [0..40]]; // Vincenzo Librandi, Aug 29 2011

Table[n^3 - n + 1, {n, 0, 40}] (* Vladimir Joseph Stephan Orlovsky, Feb 20 2012 *)
LinearRecurrence[{4,-6,4,-1},{1,1,7,25},50] (* Harvey P. Dale, Aug 17 2020 *)

a(n) = n^3 - n + 1; \\ Harry J. Smith, Jul 25 2009

G.f.: (1-3*x+9*x^2-x^3)/(1 - x)^4.  a(-n) = -A126420(n). - Bruno Berselli, Aug 29 2011
a(n) = 1 + Sum_{k=1..n} 3*(k-1)*k. - Luce ETIENNE and Michel Marcus, Nov 01 2014
E.g.f.: exp(x)*(1 + 3*x^2 + x^3). - Nikolaos Pantelidis, Feb 13 2023

Offset changed from 1 to 0 by Harry J. Smith, Jul 25 2009

A126434 Primes of the form k^6-k-1.

Original entry on oeis.org

61, 4091, 15619, 46649, 2985971, 16777199, 24137551, 63999979, 4750104199, 8303765579, 27680640569, 30840979399, 34296447191, 68719476671, 117648999929, 351298031531, 377149515539, 606355001251, 689869780961

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, A126422, A126423, A126424, A126425, A126426, A126427.

k = 6; a = {}; Do[If[PrimeQ[x^k - x - 1], AppendTo[a, x^k - x - 1]], {x, 1, 100}]; a
Select[Table[n^6-n-1,{n,200}],PrimeQ] (* Harvey P. Dale, Mar 28 2013 *)

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A126421 Numbers n for which n^3-n-1 is prime.

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

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A116581 Primes of the form k^3 - k - 1.

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

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