A091568 -id:A091568 - cp's OEIS frontend

A091567 Primes p such that p^2-p-1 is prime.

3, 5, 7, 11, 17, 29, 31, 47, 61, 67, 71, 97, 101, 127, 131, 139, 149, 181, 197, 241, 269, 307, 331, 359, 379, 397, 409, 419, 421, 449, 457, 479, 487, 491, 599, 607, 617, 619, 641, 647, 677, 709, 751, 787, 839, 857, 907, 947, 967, 977, 997, 1051, 1061, 1091

Offset: 1

T. D. Noe, Jan 21 2004

nonn

M. Cerasoli, F. Eugeni and M. Protasi, Elementi di Matematica Discreta, Bologna 1988
Emanuele Munarini and Norma Zagaglia Salvi, Matematica Discreta,UTET, CittaStudiEdizioni, Milano 1997

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

Cf. A053182 (p^2+p+1 prime), A053184 (p^2+p-1 prime), A065508 (p^2-p+1 prime).

Cf. A091568 (corresponding primes of the form p^2-p-1).

Select[Prime[Range[250]], PrimeQ[ #^2-#-1]&] (* Alexander Adamchuk, Jul 27 2006 *)

isA091567(n)=isprime(n) && isprime(n^2-n-1) \\ Michael B. Porter, May 12 2010

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)}

A119534 Largest prime divisor of numerator of the n-th Artin's product.

Original entry on oeis.org

5, 19, 41, 109, 109, 271, 271, 271, 811, 929, 929, 929, 929, 2161, 2161, 2161, 3659, 4421, 4969, 4969, 4969, 4969, 4969, 9311, 10099, 10099, 10099, 10099, 10099, 16001, 17029, 17029, 19181, 22051, 22051, 22051, 22051, 22051, 22051, 22051, 32579

Offset: 2

Alexander Adamchuk, Jul 27 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin's product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]. a(n) is prime from A091568 of the form p^2-p-1, where p is prime from A091567.

Amiram Eldar, Table of n, a(n) for n = 2..1000
Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A091568, A091567, A005596, A048296.

[Max(PrimeDivisors(Numerator(&*[1-1/(NthPrime(k)^2-NthPrime(k)):k in [1..n]]))): n in [2..45]]; // Marius A. Burtea, Feb 18 2020

Table[Max[FactorInteger[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]]]],{n,2,100}]

a(n) = Max[FactorInteger[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]]]].

A237641 Primes p of the form n^2-n-1 (for prime n) such that p^2-p-1 is also prime.

Original entry on oeis.org

5, 236681, 380071, 457651, 563249, 1441199, 1660231, 2491661, 3050261, 4106701, 5137021, 5146091, 5329171, 10617821, 15574861, 19860391, 20852921, 21349019, 21497131, 23025601, 24507449, 32495699, 36342811, 48867089, 51129649, 59082281

Offset: 1

Derek Orr, Feb 10 2014

nonn

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

These are the primes in the sequence A237527.

			5 = 3^2-3^1-1 (3 is prime) and 5^2-5-1 = 19 is prime. Since 5 is prime too, 5 is a member of this sequence.

Cf. A237527, A091567, A091568.

Select[Table[n^2-n-1,{n,Prime[Range[1000]]}],AllTrue[{#,#^2-#-1},PrimeQ]&] (* Harvey P. Dale, Aug 14 2024 *)

s=[]; forprime(n=2, 40000, 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
def poly2(x):
  if isprime(x):
    f = x**2-x-1
    if isprime(f**2-f-1):
      return True
  return False
x = 1
while x < 10**5:
  if poly2(x):
    if isprime(x**2-x-1):
      print(x**2-x-1)
  x += 1

A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

Original entry on oeis.org

1, 5, 19, 41, 109, 155, 271, 341, 505, 811, 929, 1331, 1639, 1805, 2161, 2755, 3421, 3659, 4421, 4969, 5255, 6161, 6805, 7831, 9311, 10099, 10505, 11341, 11771, 12655, 16001, 17029, 18631, 19181, 22051, 22649, 24491, 26405, 27721, 29755, 31861, 32579, 36289

Offset: 1

Kritsada Moomuang, Jan 28 2019

nonn

Terms are divisible by 5 iff p is of the form 10*m + 3 (A030431).

			a(3) = 19 because 5^2 - 5 - 1 = 19.

Robert Israel, Table of n, a(n) for n = 1..10000

Supersequence of A091568.
Subsequence of A028387 or A165900.
Second column of A378979.
Cf. A000040, A001248, A030431, A033879, A036689, A036690, A060800, A065479, A065488, A072055, A089241.
A039914 is an essentially identical sequence.

map(p -> p^2-p-1, [seq(ithprime(i),i=1..100)]); # Robert Israel, Mar 11 2019

Table[Prime[n]^2-Prime[n]-1, {n, 1, 100}] (* Jinyuan Wang, Feb 02 2019 *)

a(n) = {p=prime(n);p^2-p-1;} \\ Jinyuan Wang, Feb 02 2019

a(n) = A036689(n) - 1.
a(n) = A036690(n) - A072055(n).
a(n) = A060800(n) - A089241(n).
From Amiram Eldar, Nov 07 2022: (Start)
Product_{n>=1} (1 + 1/a(n)) = A065488.
Product_{n>=2} (1 - 1/a(n)) = A065479. (End)
a(n) = A033879(A001248(n)). [Deficiency of squares of primes] - Antti Karttunen, Dec 13 2024

A136241 Numbers n among A006093 such that n^2 + n - 1 is prime.

Original entry on oeis.org

2, 4, 6, 10, 16, 28, 30, 46, 60, 66, 70, 96, 100, 126, 130, 138, 148, 180, 196, 240, 268, 306, 330, 358, 378, 396, 408, 418, 420, 448, 456, 478, 486, 490, 598, 606, 616, 618, 640, 646, 676, 708, 750, 786, 838, 856, 906, 946, 966, 976, 996, 1050, 1060, 1090

Offset: 1

Lekraj Beedassy, Dec 23 2007

nonn

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

Cf. A136240, A091568.

Select[Prime[Range[200]]-1,PrimeQ[#^2+#-1]&] (* Harvey P. Dale, Jan 20 2019 *)

a(n) = A091567(n) - 1.

A119609 p^2-p-1 that is not prime, where p is prime.

Original entry on oeis.org

1, 155, 341, 505, 1331, 1639, 1805, 2755, 3421, 5255, 6161, 6805, 7831, 10505, 11341, 11771, 12655, 18631, 22649, 24491, 26405, 27721, 29755, 31861, 36289, 37055, 39401, 44309, 49505, 51301, 52211, 54055, 56881, 62749, 65791, 68905, 73169

Offset: 1

Alexander Adamchuk, Jul 27 2006

nonn

All prime factors of a(n) {5,11,19,29,31,41,59,61,..} belong to A038872 Primes congruent to {0, 1, 4} mod 5. Also odd primes where 5 is a square mod p. A091568 Primes of the form p^2-p-1, where p is prime. A091567 Primes p such that p^2-p-1 is prime.

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

Cf. A091568, A091567, A038872.

[q: p in PrimesUpTo(300) | IsPrime(p) and not IsPrime(q) where q is p^2 - p - 1] // Vincenzo Librandi, Sep 08 2012

lst = {}; Do[If[PrimeQ[p] && ! PrimeQ[p^2 - p - 1], AppendTo[lst, p^2 - p -1]], {p, 300}]; lst (* Vincenzo Librandi, Sep 08 2012 *)

A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

Original entry on oeis.org

3, 5, 7, 11, 13, 19, 29, 31, 41, 43, 109, 131, 157, 181, 271, 307, 379, 811, 929, 991, 1721, 1723, 2161, 2861, 3539, 3541, 3659, 4421, 4423, 4969, 5113, 6163, 6971, 8009, 8011, 9311, 10099, 10301, 10303, 10711, 16001, 17029, 17291, 17293, 19181, 19183, 22051, 22349, 22651

Offset: 1

Ralf Steiner, Aug 11 2017

nonn

This sequence contains prime chains and prime trees using an appropriate mapping form p^2 +- p +- 1 in each step, such as the chain: 3 -> 5 -> 19 -> 379 -> 143263 -> 20524143907 and the tree: 41 -> {1721, 1723}.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000040.

Union of A053183, A053185, A074268, A091568.

{p^2+(-1)^k*p+(-1)^s:p in PrimesUpTo(150), s,k in [1..2]|IsPrime(p^2+(-1)^k*p+(-1)^s)}; //  Marius A. Burtea, Nov 28 2019

select(isprime, [3,seq(op([p^2-p-1,p^2-p+1,p^2+p-1,p^2+p+1]),p=select(isprime,[seq(i,i=3..1000,2)]))]); # Robert Israel, Nov 27 2019

Select[Union[Flatten[{(#^2 + # + 1 ), (#^2 + # - 1 ), (#^2 - # + 1 ), (#^2 - # - 1 )}] &[Prime[Range[100]]]], (PrimeQ[#]) &]

A119570 Primes p such that p^2 - p - 1 is not prime.

Original entry on oeis.org

2, 13, 19, 23, 37, 41, 43, 53, 59, 73, 79, 83, 89, 103, 107, 109, 113, 137, 151, 157, 163, 167, 173, 179, 191, 193, 199, 211, 223, 227, 229, 233, 239, 251, 257, 263, 271, 277, 281, 283, 293, 311, 313, 317, 337, 347, 349, 353, 367, 373, 383, 389, 401, 431, 433

Offset: 1

Alexander Adamchuk, Jul 27 2006

nonn

Cf. A091567 (Primes p such that p^2-p-1 is prime),

A091568 (Primes of the form p^2-p-1, where p is prime).

[p: p in PrimesUpTo(700)| not IsPrime(p^2-p-1)] // Vincenzo Librandi, Jan 29 2011

Select[Prime[Range[250]], Not[PrimeQ[ #^2-#-1]]&]

A119964 Numerator of the n-th Artin product.

Original entry on oeis.org

1, 5, 19, 779, 84911, 2632241, 713337311, 1163866139, 587752400195, 476667196558145, 2856927907113011, 345688276760674331, 13819099649042566549, 4988694973304366524189, 10780569837310736058772429

Offset: 1

Alexander Adamchuk, Aug 03 2006

Artin's constant (A005596) is equal to Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,Infinity}]. n-th Artin product is Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}].

Eric Weisstein's World of Mathematics, Artin's Constant.

Cf. A119534, A091568, A091567, A005596, A048296.

Table[Numerator[Product[1-1/(Prime[k]*(Prime[k]-1)),{k,1,n}]],{n,1,20}]

a(n) = Numerator[ Product[ 1 - 1/(Prime[k]*(Prime[k]-1)), {k,1,n}]].

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A091567 Primes p such that p^2-p-1 is prime.

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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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A119534 Largest prime divisor of numerator of the n-th Artin's product.

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A237641 Primes p of the form n^2-n-1 (for prime n) such that p^2-p-1 is also prime.

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A306190 a(n) = p^2 - p - 1 where p = prime(n), the n-th prime.

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A136241 Numbers n among A006093 such that n^2 + n - 1 is prime.

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A119609 p^2-p-1 that is not prime, where p is prime.

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Magma

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A290817 Primes of at least one of the forms p^2 +- p +- 1, where p is prime.

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