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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.

%I A237640 #13 May 22 2025 10:21:36
%S A237640 122,340352,830519696,11479086422,266390469692,310503441398,
%T A237640 2718130415306,14837993872846,59538248604388,889257663626476,
%U A237640 2496623039993996,6427431330617746,7120028814392596,10777302002014868,12942591289426088,24039736320940828
%N A237640 Numbers n of the form p^5 - Phi_5(p) (for prime p) such that n^5 - Phi_5(n) is also prime.
%C A237640 All numbers are congruent to 2 mod 10, 6 mod 10, or 8 mod 10.
%C A237640 x^5 - Phi_5(x) = x^5-x^4-x^3-x^2-x-1.
%e A237640 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.
%o A237640 (Python)
%o A237640 import sympy
%o A237640 from sympy import isprime
%o A237640 def poly5(x):
%o A237640   if isprime(x):
%o A237640     f = x**5-x**4-x**3-x**2-x-1
%o A237640     if isprime(f**5-f**4-f**3-f**2-f-1):
%o A237640       return True
%o A237640   return False
%o A237640 x = 1
%o A237640 while x < 10**5:
%o A237640   if poly5(x):
%o A237640     print(x**5-x**4-x**3-x**2-x-1)
%o A237640   x += 1
%Y A237640 Cf. A125083, A237527, A237528, A237639.
%K A237640 nonn
%O A237640 1,1
%A A237640 _Derek Orr_, Feb 10 2014