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A086598
Number of distinct prime factors in Lucas(n).
7
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
1,6
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
T. D. Noe, Table of n, a(n) for n = 1..1000 (using Blair Kelly's data)
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).
Sequence in context: A368542 A344234 A338912 * A211261 A344174 A336431
KEYWORD
hard,nonn
AUTHOR
T. D. Noe, Jul 24 2003
STATUS
approved