A237527 -id:A237527 - cp's OEIS frontend

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

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

A230026 Primes p such that f(f(p)) is prime, where f(n) = n^2-n-1 = A165900(n).

Original entry on oeis.org

3, 13, 23, 53, 59, 83, 107, 167, 173, 179, 211, 223, 229, 257, 317, 349, 367, 431, 443, 487, 503, 509, 541, 571, 613, 617, 673, 677, 683, 751, 823, 1021, 1031, 1093, 1103, 1109, 1123, 1201, 1231, 1289, 1301, 1319, 1361, 1373, 1427, 1451

Offset: 1

Derek Orr, Feb 23 2014

nonn

Note that f(f(f(n))) = (-1 + 4*n - 3*n^3 + n^4)*(1 + n - 3*n^2 - n^3 + n^4) is always composite. - Zak Seidov, Nov 10 2014

			3 is prime and (3^2-3-1)^2-(3^2-3-1)-1 = 19 is also prime. So, 3 is a member of this sequence.

Cf. A000040, A237527, A002327.

import sympy
from sympy import isprime
def f(x):
    return x**2-x-1
{p for p in range(10**4) if isprime(p) and isprime(f(f(p)))}

f = lambda x: x^2-x-1
[p for p in primes(1452) if is_prime(f(f(p)))] # Peter Luschny, Mar 02 2014

A237527(n) = A165900(a(n)). - M. F. Hasler, Mar 01 2014

A237640 Numbers n of the form p^5 - Phi_5(p) (for prime p) such that n^5 - Phi_5(n) is also prime.

Original entry on oeis.org

122, 340352, 830519696, 11479086422, 266390469692, 310503441398, 2718130415306, 14837993872846, 59538248604388, 889257663626476, 2496623039993996, 6427431330617746, 7120028814392596, 10777302002014868, 12942591289426088, 24039736320940828

Offset: 1

Derek Orr, Feb 10 2014

nonn

All numbers are congruent to 2 mod 10, 6 mod 10, or 8 mod 10.

x^5 - Phi_5(x) = x^5-x^4-x^3-x^2-x-1.

			122 = 3^5-3^4-3^3-3^2-3^1-1 (3 is prime) and 122^5-122^4-122^3-122^2-122^1-1 = 26803717321 is prime. Thus, 122 is a member of this sequence.

Cf. A125083, A237527, A237528, A237639.

import sympy
from sympy import isprime
def poly5(x):
  if isprime(x):
    f = x**5-x**4-x**3-x**2-x-1
    if isprime(f**5-f**4-f**3-f**2-f-1):
      return True
  return False
x = 1
while x < 10**5:
  if poly5(x):
    print(x**5-x**4-x**3-x**2-x-1)
  x += 1

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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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A230026 Primes p such that f(f(p)) is prime, where f(n) = n^2-n-1 = A165900(n).

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A237640 Numbers n of the form p^5 - Phi_5(p) (for prime p) such that n^5 - Phi_5(n) is also prime.

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