A002327 -id:A002327 - cp's OEIS frontend

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

-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

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 *)

A237615 a(n) = |{0 < k < n: k^2 + k - 1 and pi(k*n) are both prime}|, where pi(.) is given by A000720.

Original entry on oeis.org

0, 0, 1, 1, 0, 2, 2, 1, 2, 1, 3, 2, 1, 4, 1, 3, 4, 4, 2, 4, 3, 6, 2, 2, 2, 3, 7, 4, 3, 4, 5, 6, 1, 3, 2, 3, 9, 3, 3, 4, 7, 5, 8, 5, 2, 2, 5, 5, 4, 5, 6, 4, 5, 6, 10, 6, 6, 10, 9, 9, 10, 12, 2, 8, 7, 3, 6, 6, 4, 6

Offset: 1

Zhi-Wei Sun, Feb 10 2014

nonn

Conjecture: (i) a(n) > 0 for all n > 5.
(ii) For each n = 4, 5, ..., there is a positive integer k < n with k^2 + k - 1 and pi(k*n) + 1 both prime. Also, for any integer n > 6, there is a positive integer k < n with k^2 + k - 1 and pi(k*n) - 1 both prime.
(iii) For every integer n > 15, there is a positive integer k < n such that pi(k) - 1 and pi(k*n) are both prime.
Note that part (i) is a refinement of the first assertion in the comments in A237578.

			a(8) = 1 since 4^2 + 4 - 1 = 19 and pi(4*8) = 11 are both prime.
a(33) = 1 since 28^2 + 28 - 1 = 811 and pi(28*33) = 157 are both prime.

Zhi-Wei Sun, Table of n, a(n) for n = 1..5000
Zhi-Wei Sun, A combinatorial conjecture on primes, a message to Number Theory List, Feb. 9, 2014.

Cf. A000040, A000720, A002327, A045546, A237578, A237597, A237598.

p[k_,n_]:=PrimeQ[k^2+k-1]&&PrimeQ[PrimePi[k*n]]
a[n_]:=Sum[If[p[k,n],1,0],{k,1,n-1}]
Table[a[n],{n,1,70}]

A237642 Primes of the form n^2-n-1 (for some n) such that p^2-p-1 is also prime.

Original entry on oeis.org

5, 11, 29, 71, 131, 181, 379, 419, 599, 1979, 2069, 3191, 4159, 13339, 14519, 17291, 19739, 20879, 21169, 26731, 30449, 31151, 39799, 48619, 69959, 70489, 112559, 122849, 132859, 139501, 149381, 183611, 186191, 198469, 212981, 222311, 236681

Offset: 1

Derek Orr, Feb 10 2014

nonn

Except a(1), all numbers are congruent to 1 mod 10 or 9 mod 10.

			11 is prime and equals 4^2-4-1, and 11^2-11-1 = 109 is prime. So, 11 is a member of this sequence.

Harvey P. Dale, Table of n, a(n) for n = 1..1000

Cf. A002327, A091568.

Select[Table[n^2-n-1,{n,500}],AllTrue[{#,#^2-#-1},PrimeQ]&] (* Harvey P. Dale, Feb 27 2023 *)

s=[]; for(n=1, 1000, p=n^2-n-1; if(isprime(p) && isprime(p^2-p-1), s=concat(s, p))); s \\ Colin Barker, Feb 11 2014

import sympy
from sympy import isprime
{print(n**2-n-1) for n in range(10**3) if isprime(n**2-n-1) and isprime((n**2-n-1)**2-(n**2-n-1)-1)}

A356247 Denominator of the continued fraction 1/(2 - 3/(3 - 4/(4 - 5/(...(n-1) - n/(-1))))).

Original entry on oeis.org

1, 5, 11, 19, 29, 41, 11, 71, 89, 109, 131, 31, 181, 19, 239, 271, 61, 31, 379, 419, 461, 101, 29, 599, 59, 701, 151, 811, 79, 929, 991, 211, 59, 41, 1259, 1, 281, 1481, 1559, 149, 1721, 1, 61, 1979, 2069, 2161, 1, 2351, 79, 2549, 241, 1, 2861, 2969, 3079, 3191

Offset: 2

Mohammed Bouras, Jul 30 2022

Conjecture 1: Every term of this sequence is either a prime number or 1.
Conjecture 2: The sequence contains all prime numbers which ends with a 1 or 9.
Conjecture 3: Except for 5, the primes all appear exactly twice.
a(n) divides n^2 - n - 1, which is the unreduced denominator.
Conjecture: The ordered sequence of prime values is A038872. - Bill McEachen, Jul 28 2025
For a proof of Conjectures 1-3, see Cloitre (2025). - Sean A. Irvine, Aug 25 2025

			For n=2, 1/(2 - 3) = -1, so a(2)=1.
For n=3, 1/(2 - 3/(3 - 4)) = 1/5, so a(3)=5.
For n=4, 1/(2 - 3/(3 - 4/(4 - 5))) = 7/11, so a(4)=11.
For n=5, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6)))) = 23/19, so a(5)=19.
For n=6, 1/(2 - 3/(3 - 4/(4 - 5/(5 - 6/(6 - 7))))) = 73/29, so a(6)=29.
a(23) = a(79) = 23 + 79 - 1 = 101.
a(26) = a(34) = gcd(26^2 - 26 -1, 34^2 - 34 - 1) = gcd(649, 1121) = 59.

Michael De Vlieger, Table of n, a(n) for n = 2..10001
Mohammed Bouras, A new sequence of prime numbers, Romanian Math. Mag. (15 August 2022). See Table 2 p. 2.
Mohammed Bouras, A New Sequence of Prime Numbers, EasyChair Preprint 8686, 2022.
Mohammed Bouras, The Distribution Of Prime Numbers And Continued Fractions, (ppt) (2022).
Benoit Cloitre, On the sequence A356247, 2025.

Cf. A002327 (primes of the form k^2-k-1), A028387, A051403, A165900, A356684.

a[n_] := ContinuedFractionK[-i-1, If[i == n, 1, i+1], {i, 1, n}] //
   Denominator;
Table[a[n], {n, 2, 100}] (* Jean-François Alcover, Aug 11 2022 *)

a(n) = if (n==1, 1, n--; my(v = vector(2*n, k, (k+4)\2)); my(q = 1/(v[2*n-1] - v[2*n])); forstep(k=2*n-3, 1, -2, q = v[k] - v[k+1]/q; ); denominator(1/q)); \\ Michel Marcus, Aug 07 2022

from fractions import Fraction
def A356247(n):
    k = -1
    for i in range(n-1,1,-1):
        k = i-Fraction(i+1,k)
    return abs(k.numerator) # Chai Wah Wu, Aug 23 2022

a(n) = (n^2 - n - 1)/gcd(n^2 - n - 1, A356684(n)).
If conjecture 3 is true, then we have:
   a(n) = a(m) = n + m - 1.
   a(n) = a(m) = gcd(n^2 - n - 1, m^2 - m - 1).
   a(n) = a(a(n) - n + 1).

A076846 Primes of the form n^k + n - 1, where k>0 is minimal.

Original entry on oeis.org

3, 5, 7, 29, 11, 13, 71, 17, 19, 131, 23, 181, 2177953337809371149, 29, 31, 83537, 5849, 37, 419, 41, 43, 279863, 47, 15649, 701, 53, 811, 420707233300229, 59, 61

Offset: 2

Benoit Cloitre, Nov 20 2002

nonn

The next term is too large to include.

Reinhard Zumkeller, Table of n, a(n) for n = 2..100

Cf. A076845, A078179, A010051, A002327.

a076846 n = n ^ (a076845 n) + n - 1
-- Reinhard Zumkeller, Jul 17 2014

Offset corrected by Reinhard Zumkeller, Jul 17 2014

cp's OEIS Frontend

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

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

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

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A237615 a(n) = |{0 < k < n: k^2 + k - 1 and pi(k*n) are both prime}|, where pi(.) is given by A000720.

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A237642 Primes of the form n^2-n-1 (for some n) such that p^2-p-1 is also prime.

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