A091567 -id:A091567 - cp's OEIS frontend

A053182 Primes p such that p^2 + p + 1 is prime.

2, 3, 5, 17, 41, 59, 71, 89, 101, 131, 167, 173, 293, 383, 677, 701, 743, 761, 773, 827, 839, 857, 911, 1091, 1097, 1163, 1181, 1193, 1217, 1373, 1427, 1487, 1559, 1583, 1709, 1811, 1847, 1931, 1973, 2129, 2273, 2309, 2339, 2411, 2663, 2729, 2789, 2957

Offset: 1

Enoch Haga, Mar 01 2000

Roger Horn computed the first 776 terms of this sequence around 1961 to test (with Paul Bateman) their conjecture on the density of simultaneous primes in polynomials. - Charles R Greathouse IV, Apr 05 2011
Starting with a(3)=5 all terms are of the form 6k-1, k in A147683. - Zak Seidov, Nov 10 2008
Primes p such that the sum of divisors of p^2 (sigma(p^2) = A000203(p^2) = p^2+p+1) is prime. - Claudio Meller, Apr 07 2011
The generated prime numbers p^2 + p + 1 are exactly A053183. - Bernard Schott, Dec 20 2012
Positive squarefree k such that the sum of divisors of k^2 is prime. - Peter Munn, Feb 02 2018

Jon E. Schoenfield, Table of n, a(n) for n = 1..10000 (first 2650 terms from M. F. Hasler).
Paul T. Bateman and Roger A. Horn, A heuristic asymptotic formula concerning the distribution of prime numbers, Math. Comp. 16 (1962), 363-367.
Paolo Santonastaso and Ferdinando Zullo, Linearized trinomials with maximum kernel, arXiv:2012.14861 [math.NT], 2020.

Cf. A053184, A065508, A091567, A147683, A188596.

[p: p in PrimesUpTo(10000) | IsPrime(p^2+p+1)]; // Vincenzo Librandi, Aug 06 2010

Select[Prime[Range[427]], PrimeQ[#^2+#+1]&] (* Bruno Berselli, Nov 08 2011 *)

isA053182(n)=isprime(n) && isprime(n^2+n+1)  \\ Michael B. Porter, Apr 23 2010

c=0; forprime(p=1,default(primelimit), isprime(p^2+p+1) & write("/tmp/b053182.txt",c++," "p))  \\ M. F. Hasler, Apr 07 2011

List changed to cross-reference by Franklin T. Adams-Watters, May 12 2010

A053184 Primes p such that p^2+p-1 is prime.

Original entry on oeis.org

2, 3, 5, 11, 13, 19, 31, 41, 53, 59, 83, 89, 101, 103, 131, 149, 163, 181, 191, 193, 199, 233, 241, 263, 281, 331, 349, 373, 401, 419, 431, 433, 449, 461, 463, 499, 541, 569, 571, 659, 673, 683, 691, 709, 739, 743, 761, 769, 809, 863, 881, 941, 1013, 1039

Offset: 1

Enoch Haga, Mar 01 2000

Pierre CAMI, Table of n, a(n) for n = 1..10000

Cf. A053182, A065508, A091567.

[p: p in PrimesUpTo(1050) | IsPrime(p^2+p-1)]; // Bruno Berselli, Nov 08 2011

Select[Prime[Range[175]], PrimeQ[#^2+#-1]&] (* Bruno Berselli, Nov 08 2011 *)

isA053184(n)=isprime(n) && isprime(n^2+n-1) \\ Michael B. Porter, May 10 2010

A065508 Primes p such that p^2 - p + 1 is prime.

Original entry on oeis.org

2, 3, 7, 13, 67, 79, 139, 151, 163, 193, 337, 349, 379, 457, 541, 613, 643, 727, 769, 919, 991, 1021, 1093, 1117, 1201, 1231, 1381, 1423, 1549, 1567, 1597, 1621, 1693, 1747, 1789, 1801, 1933, 1987, 2011, 2017, 2113, 2137, 2143, 2239, 2281, 2557, 2647

Offset: 1

Vladeta Jovovic, Nov 26 2001

The primes p^2 - p + 1 are in A074268.

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

Cf. A053182, A053184, A091567, A074268.

[n: n in [0..10000]| IsPrime(n) and IsPrime(n^2 - n + 1)] // Vincenzo Librandi, Aug 07 2010

Select[Prime[Range[500]],PrimeQ[#^2-#+1]&] (* Harvey P. Dale, Oct 06 2015 *)

{ n=0; for (m=1, 10^9, p=prime(m); if (isprime(p^2 - p + 1), write("b065508.txt", n++, " ", p); if (n==1000, return)) ) } \\ Harry J. Smith, Oct 20 2009

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

Original entry on oeis.org

5, 19, 41, 109, 271, 811, 929, 2161, 3659, 4421, 4969, 9311, 10099, 16001, 17029, 19181, 22051, 32579, 38611, 57839, 72091, 93941, 109229, 128521, 143261, 157211, 166871, 175141, 176819, 201151, 208391, 228961, 236681, 240589, 358201, 367841

Offset: 1

T. D. Noe, Jan 21 2004

nonn

Vincenzo Librandi, Table of n, a(n) for n = 1..5000

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

lst={};Do[p=Prime[n];p=p^2-p-1;If[PrimeQ[p],AppendTo[lst,p]],{n,3*5!}];lst (* Vladimir Joseph Stephan Orlovsky, Jan 27 2009 *)
Select[#^2-#-1&/@Prime[Range[200]],PrimeQ] (* Harvey P. Dale, May 15 2016 *)

isA091568(n)={local(i);i=floor((1/2)+sqrt(n+(5/4)));isprime(i) && n==i^2-i-1 && isprime(n)} \\ Michael B. Porter, May 12 2010

A154939 Primes p such that (p-1)*(p+1)-+p are primes.

Original entry on oeis.org

3, 5, 11, 31, 101, 131, 149, 181, 241, 331, 419, 449, 709, 1051, 1061, 1171, 1409, 1549, 1579, 1699, 1759, 1831, 2069, 3229, 3449, 3761, 3911, 4159, 4951, 5821, 6029, 6481, 6661, 6679, 6899, 7079, 7151, 7229, 7369, 8101, 8219, 8629, 8861, 9091, 9161, 9521

Offset: 1

Vladimir Joseph Stephan Orlovsky, Jan 17 2009

nonn

That is, primes p such that p^2+p-1 and p^2-p-1 are both primes: intersection of A053184 and A091567. - Michel Marcus, Jul 10 2016

			2*4=8-+3 -> primes, 4*6=24-+5 -> primes,...

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

Cf. A053184, A038872, A141158, A091567, A098058, A038936, A089270, A140559.

[p: p in PrimesUpTo(10000) | IsPrime(p^2+p-1) and IsPrime(p^2-p-1)]; // Vincenzo Librandi, Jul 10 2016

lst={};Do[p=Prime[n];If[PrimeQ[(p-1)*(p+1)-p]&&PrimeQ[(p-1)*(p+1)+p],AppendTo[lst,p]],{n,7!}];lst
Select[Prime[Range[1500]], And@@PrimeQ/@{#^2 - # - 1, #^2 + # - 1} &] (* Vincenzo Librandi, Jul 10 2016 *)
Select[Prime[Range[1500]],AllTrue[(#-1)(#+1)+{#,-#},PrimeQ]&] (* Harvey P. Dale, Sep 21 2023 *)

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

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.

A243471 Primes p such that p^6 - p^5 + 1 is prime.

Original entry on oeis.org

3, 31, 73, 181, 367, 373, 523, 631, 733, 1021, 1039, 1171, 1489, 1723, 1777, 2203, 2557, 2683, 3121, 3187, 3319, 4441, 4591, 4621, 4801, 4957, 5113, 5167, 5323, 5431, 5659, 5839, 5851, 5857, 6883, 7057, 7129, 7297, 7309, 7477, 7993, 8017, 8209, 8221, 8689, 8821

Offset: 1

K. D. Bajpai, Jun 05 2014

nonn

			31 appears in the sequence because it is prime and 31^6 - 31^5  + 1 = 858874531 is also prime.
73 appears in the sequence because it is prime and 73^6 - 73^5  + 1 = 149261154697 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..5457

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

A243471 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5+1; if isprime (b) then RETURN (a); fi; end: seq(A243471 (), n=1..2000);

c=0; Do[k=Prime[n]; If[PrimeQ[k^6-k^5+1], c++; Print[c," ",k]], {n,1,200000}];

A243472 Primes p such that p^6 - p^5 - 1 is prime.

Original entry on oeis.org

2, 31, 101, 151, 181, 199, 229, 277, 307, 317, 379, 439, 479, 491, 647, 691, 797, 911, 997, 1039, 1051, 1181, 1291, 1367, 1381, 1471, 1511, 1549, 1657, 1709, 1847, 1867, 1987, 2081, 2099, 2111, 2207, 2467, 2621, 2707, 3041, 3221, 3259, 3541, 3571, 3581, 3769

Offset: 1

K. D. Bajpai, Jun 05 2014

nonn

			31 appears in the sequence because it is prime and 31^6 - 31^5 - 1 = 858874529 is also prime.
101 appears in the sequence because it is prime and 101^6 - 101^5  - 1 = 1051010050099 is also prime.

K. D. Bajpai, Table of n, a(n) for n = 1..4699

Cf. A000040, A091567, A053182, A053183, A053184, A238136.

A243472 := proc() local a, b; a:=ithprime(n); b:= a^6-a^5-1; if isprime (b) then RETURN (a); fi; end: seq(A243472 (), n=1..2000);

c = 0;  Do[k=Prime[n]; If[PrimeQ[k^6-k^5-1], c++; Print[c," ",k]], {n,1,200000}];
Select[Prime[Range[600]],PrimeQ[#^6-#^5-1]&] (* Harvey P. Dale, Jan 21 2015 *)

s=[]; forprime(p=2, 4000, if(isprime(p^6-p^5-1), s=concat(s, p))); s \\ Colin Barker, Jun 06 2014

cp's OEIS Frontend

A053182 Primes p such that p^2 + p + 1 is prime.

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A053184 Primes p such that p^2+p-1 is prime.

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A065508 Primes p such that p^2 - p + 1 is prime.

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

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A154939 Primes p such that (p-1)*(p+1)-+p are primes.

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

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