A230026 -id:A230026 - cp's OEIS frontend

A237527 Numbers n of the form p^2-p-1 = A165900(p), for prime p, such that n^2-n-1 = A165900(n) is also prime.

5, 155, 505, 2755, 3421, 6805, 11341, 27721, 29755, 31861, 44309, 49505, 52211, 65791, 100171, 121451, 134321, 185329, 195805, 236681, 252505, 258571, 292139, 325469, 375155, 380071, 452255, 457651, 465805, 563249, 676505, 1041419, 1061929

Offset: 1

Derek Orr, Feb 09 2014

nonn

All numbers are congruent to 1 mod 10, 5 mod 10, or 9 mod 10.

A subsequence of A165900 and A028387. - M. F. Hasler, Mar 01 2014

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

Cf. A237360, A039914, A002327.

s=[]; forprime(p=2, 40000, n=p^2-p-1; if(isprime(n^2-n-1), s=concat(s, n))); s \\ Colin Barker, Feb 10 2014

import sympy
from sympy import isprime
{print(n**2-n-1) for n in range(10**4) if isprime(n) and isprime((n**2-n-1)**2-(n**2-n-1)-1)}

a(n) = A165900(A230026(n)). - M. F. Hasler, Mar 01 2014

A238447 Primes p such that f(p) and f(f(p)) are both prime, where f(x) = x^2-x-1.

Original entry on oeis.org

3, 487, 617, 677, 751, 1201, 1289, 1579, 1747, 2027, 2267, 2269, 2309, 3259, 3947, 4457, 4567, 4621, 4637, 4799, 4951, 5701, 6029, 6991, 7151, 7687, 7867, 9187, 9209, 9341, 9587, 9829, 11321, 12301, 12541, 12781, 13177, 13649, 15919, 16349

Offset: 1

Derek Orr, Feb 26 2014

nonn

Intersection of A230026 and A091567.

Note that f(f(f(p))) is always composite. - Zak Seidov, Nov 10 2014

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

Zak Seidov, Table of n, a(n) for n = 1..1000

Cf. A237642, A230026, A091567.

Select[Prime[Range[2000]],AllTrue[Rest[NestList[#^2-#-1&,#,2]],PrimeQ]&] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Oct 03 2017 *)

import sympy
from sympy import isprime
def f(x):
  return x**2-x-1
{print(p) for p in range(10**5) if isprime(p) and isprime(f(p)) and isprime(f(f(p)))}

cp's OEIS Frontend

A237527 Numbers n of the form p^2-p-1 = A165900(p), for prime p, such that n^2-n-1 = A165900(n) is also prime.

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