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

0,7

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

Formula

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

Mathematica

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

Prog

(PARI) a(n)=omega(fibonacci(n-1)+fibonacci(n+1)) \\ Charles R Greathouse IV, Sep 14 2015
(Magma) [#PrimeDivisors(Lucas(n)): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

Crossrefs

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

Cf. A053027, A053028, A053032.

Sequence in context: A368542 A344234 A338912 * A211261 A344174 A336431

Adjacent sequences: A086595 A086596 A086597 * A086599 A086600 A086601

Keyword

hard,nonn

Author

T. D. Noe, Jul 24 2003

Extensions

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

Status

approved