A237528 -id:A237528 - cp's OEIS frontend

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

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

A230027 Primes p such that f(f(p)) is prime, where f(x) = x^3-x^2-x-1.

Original entry on oeis.org

29, 53, 79, 83, 149, 167, 193, 227, 283, 311, 317, 401, 709, 907, 953, 1093, 1327, 1427, 1511, 1579, 1613, 1663, 1801, 1901, 1987, 2027, 2029, 2293, 2341, 2741, 2887, 3083, 3163, 3229, 3329, 3511, 3733, 4007, 4127, 4153, 4337, 4789, 5531

Offset: 1

Derek Orr, Feb 23 2014

nonn

			29 is prime and (29^3-29^2-29-1)^3 - (29^3-29^2-29-1)^2 - (29^3-29^2-29-1) - 1 = 13007166227989 is prime. Thus, 29 is a member of this sequence.

Cf. A000040, A237528, A162295.

f[n_] := n^3 - n^2 - n - 1; Select[ Prime@ Range[2, 740], PrimeQ@ f@ f@# &] (* Robert G. Wilson v, Mar 07 2014 *)

from sympy import isprime
def f(x):
  return x**3-x**2-x-1
{print(p) for p in range(10**4) if isprime(p) and isprime(f(f(p)))}

cp's OEIS Frontend

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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