A237639 Numbers n = p^4-p^3-p^2-p-1 (for prime p) such that n^4-n^3-n^2-n-1 is prime.

41, 56133395601, 89362058601, 590884122501, 1275627652881, 2775672202617, 6212311361721, 7534036143501, 27344792789601, 61180709716101, 124857759197601, 206926840439901, 580608824590341, 603653936046501, 1442441423278281, 1864059458505657

Offset: 1

Author

Derek Orr, Feb 10 2014

Keywords

• nonn

Comments

All numbers are congruent to 1 mod 10 or 7 mod 10.

41 is the only prime in the sequence, since one of p, n, and n^4-n^3-n^2-n-1 must be divisible by 3. - Charles R Greathouse IV, Feb 11 2014

Examples

			41 = 3^4-3^3-3^2-3^1-1 (3 is prime) and 41^4-41^3-41^2-41^1-1 = 2755117 is prime. So, 41 is a member of this sequence.
		

Crossrefs

Cf. A125082.

Programs

PARI

s=[]; forprime(p=2, 7000, n=p^4-p^3-p^2-p-1; if(isprime(n^4-n^3-n^2-n-1), s=concat(s, n))); s \\ Colin Barker, Feb 11 2014

Python

import sympy
from sympy import isprime
def poly4(x):
  if isprime(x):
    f = x**4-x**3-x**2-x-1
    if isprime(f**4-f**3-f**2-f-1):
      return True
  return False
x = 1
while x < 10**5:
  if poly4(x):
    print(x**4-x**3-x**2-x-1)
  x += 1