A237361 Numbers n of the form n = Phi_5(p) (for prime p) such that Phi_5(n) is also prime.

4435770414505, 30562950873505, 32152890387805, 60700878873905, 936037312559305, 1279875801783805, 3780430049614405, 6055088920612205, 10370026462436905, 12160851727605005, 16956369914710105, 18746881534017005, 20813869508536105, 30740855019988405

Offset: 1

Author

Derek Orr, Feb 06 2014

Keywords

• nonn

Comments

Phi_5(x) = x^4 + x^3 + x^2 + x + 1 is the fifth cyclotomic polynomial, see A053699.

All numbers are congruent to 5 mod 100.

The definition requires p to be prime, Phi_5(p) does not need to be prime, but Phi_5(Phi_5(p)) must be prime.

Examples

			4435770414505 = 1451^4+1451^3+1451^2+1451+1 (1451 is prime), and 4435770414505^4+4435770414505^3+4435770414505^2+4435770414505+1 = 387147304469214558406348338836395337085545589397781 is prime. Thus, 4435770414505 is a member of this sequence.
		

Crossrefs

Cf. A131992, A088548.

Programs

PARI

forprime(p=2,1e7, k=polcyclo(5,p) ; if( ispseudoprime(polcyclo(5,k)), print1(k", "))) \\ Charles R Greathouse IV, Feb 07 2014

Python

import sympy
from sympy import isprime
{print(n**4+n**3+n**2+n+1) for n in range(10**5) if isprime(n) and isprime((n**4+n**3+n**2+n+1)**4+(n**4+n**3+n**2+n+1)**3+(n**4+n**3+n**2+n+1)**2+(n**4+n**3+n**2+n+1)+1)}