A086598 Number of distinct prime factors in Lucas(n).

1, 0, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 2, 3, 1, 1, 3, 1, 2, 3, 3, 2, 3, 3, 2, 3, 2, 2, 4, 1, 2, 3, 3, 4, 4, 1, 2, 4, 3, 1, 5, 2, 4, 6, 3, 1, 4, 2, 4, 4, 3, 1, 4, 4, 2, 4, 3, 3, 6, 1, 2, 6, 2, 5, 5, 2, 2, 5, 4, 1, 4, 2, 3, 7, 2, 4, 4, 1, 2, 5, 4, 2, 6, 4, 2, 5, 3, 2, 6, 3, 3, 4, 4, 5, 4, 2, 4, 7, 4, 3, 6, 3, 4, 9

Offset: 0

Author

T. D. Noe, Jul 24 2003

Keywords

• hard
• nonn

Comments

Interestingly, the Lucas numbers separate the primes into three disjoint sets: (A053028) primes that do not divide any Lucas number, (A053027) primes that divide Lucas numbers of even index and (A053032) primes that divide Lucas numbers of odd index.

Links

• Max Alekseyev, Table of n, a(n) for n = 0..1411 (first 1000 terms from T. D. Noe)
• Blair Kelly, Fibonacci and Lucas Factorizations
• Eric Weisstein's World of Mathematics, Lucas Number

Crossrefs

Cf. A000204 (Lucas numbers), A086599 (number of prime factors, counting multiplicity), A086600 (number of primitive prime factors).

Cf. A053027, A053028, A053032.

Programs

Magma

[#PrimeDivisors(Lucas(n)): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

Mathematica

Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; Table[Length[FactorInteger[Lucas[n]]], {n, 150}]

PARI

a(n)=omega(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Sep 14 2015

Formula

a(n) = Sum{d|n and n/d odd} A086600(d) + 1 if 6|n, a Mobius-like transform

Extensions

a(0)=1 prepended by Max Alekseyev, Jun 15 2025