A237446 Primes p such that Phi_7(Phi_7(p)) is prime.

29, 197, 673, 2297, 3613, 5923, 6133, 6917, 8219, 13553, 15667, 17137, 21911, 30941, 33587, 35407, 38053, 44017, 45557, 46663, 51241, 53453, 65731, 67187, 82349, 94151, 115361, 132287, 143711, 164011, 164291, 165523, 178613, 180797, 182141, 187139, 188609, 192977

Offset

1,1

Comments

Phi_k(x) is the k-th cyclotomic polynomial, see A013595 or A013596.

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Wikipedia, Cyclotomic polynomial

Formula

Phi_7(x) = x^6 + x^5 + x^4 + x^3 + x^2 + x + 1. - Georg Fischer, Mar 07 2026

Example

29 is prime and f(29^6+29^5+29^4+29^3+29^2+29+1) = 54672347801779330810964871392077416495507203132755717 is prime. Thus, 29 is a member of this sequence.

Maple

f:= unapply(numtheory[cyclotomic](7, x), x):
q:= p-> isprime(p) and isprime(f(f(p))):
select(q, [$1..183000])[];  # Alois P. Heinz, Mar 07 2026

Prog

(Python)
import sympy
from sympy import isprime
{print(n) for n in range(10**6) if isprime(n) and isprime((n**6+n**5+n**4+n**3+n**2+n+1)**6+(n**6+n**5+n**4+n**3+n**2+n+1)**5+(n**6+n**5+n**4+n**3+n**2+n+1)**4+(n**6+n**5+n**4+n**3+n**2+n+1)**3+(n**6+n**5+n**4+n**3+n**2+n+1)**2+(n**6+n**5+n**4+n**3+n**2+n+1)+1)}

Crossrefs

Cf. A053716, A088550, A163268, A237364.

Sequence in context: A142226 A323576 A142653 * A144731 A200471 A027724

Adjacent sequences: A237443 A237444 A237445 * A237447 A237448 A237449

Keyword

nonn,changed

Author

Derek Orr, Feb 08 2014

Extensions

Definition corrected by Georg Fischer, Mar 07 2026

Status

approved