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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.

%I A237639 #16 May 22 2025 10:21:36
%S A237639 41,56133395601,89362058601,590884122501,1275627652881,2775672202617,
%T A237639 6212311361721,7534036143501,27344792789601,61180709716101,
%U A237639 124857759197601,206926840439901,580608824590341,603653936046501,1442441423278281,1864059458505657
%N 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.
%C A237639 All numbers are congruent to 1 mod 10 or 7 mod 10.
%C A237639 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
%e A237639 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.
%o A237639 (Python)
%o A237639 import sympy
%o A237639 from sympy import isprime
%o A237639 def poly4(x):
%o A237639   if isprime(x):
%o A237639     f = x**4-x**3-x**2-x-1
%o A237639     if isprime(f**4-f**3-f**2-f-1):
%o A237639       return True
%o A237639   return False
%o A237639 x = 1
%o A237639 while x < 10**5:
%o A237639   if poly4(x):
%o A237639     print(x**4-x**3-x**2-x-1)
%o A237639   x += 1
%o A237639 (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
%Y A237639 Cf. A125082.
%K A237639 nonn
%O A237639 1,1
%A A237639 _Derek Orr_, Feb 10 2014