A373194 Numbers k such that phi(k) is a Lucas number.

1, 2, 3, 4, 5, 6, 8, 10, 12, 19, 27, 38, 54, 2049, 2732, 4098, 5779, 11558, 36717, 48956, 73434, 21994424093409, 29325898791212, 43988848186818, 439894502304193355596420713117, 586526003072257807461894284156, 879789004608386711192841426234, 56570478046795035524653081529155199270281, 56570478046795035532692004624509431078281

Offset: 1

Author

Darío Clavijo, May 27 2024

Keywords

• nonn

Links

• Hugo Pfoertner, Table of n, a(n) for n = 1..64
• Max Alekseyev, PARI/GP Scripts for Miscellaneous Math Problems (invphi.gp).

Crossrefs

Cf. A000010, A000032, A102460, A280592.

Programs

Mathematica

lucasQ[n_] := Or @@ (IntegerQ[Sqrt[#]] & /@ (5*n^2 + 20*{-1, 1})); Select[Range[10^4], lucasQ[EulerPhi[#]] &] (* Amiram Eldar, May 27 2024 *)

PARI

isok(k) = islucas(eulerphi(k)); \\ using islucas from A102460 \\ Michel Marcus, May 27 2024

PARI

\\ read Max Alekseyev's invphi.gp
a373194(uptoNLucas) = my(A=List()); for(n=0, uptoNLucas, my(L = invphi(fibonacci(n+1) + fibonacci(n-1))); if(#L, for(k=1, #L, listput(A,L[k])))); Set(A);
a373194(150) \\ Hugo Pfoertner, Jun 10 2024

Python

from sympy.ntheory.primetest import is_square
from sympy import totient
islucas = lambda n: is_square(5*n*n - 20) or is_square(5*n*n + 20)
print([n for n in range(1,10**4) if islucas(totient(n))])

Extensions

a(18)-a(21) from Amiram Eldar, May 27 2024

a(22) onwards from Hugo Pfoertner, May 27 2024