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%I #16 Sep 08 2022 08:45:11
%S 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,
%T 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,
%U 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
%N Number of distinct prime factors in Lucas(n).
%C 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.
%H T. D. Noe, <a href="/A086598/b086598.txt">Table of n, a(n) for n = 1..1000</a> (using Blair Kelly's data)
%H Blair Kelly, <a href="http://mersennus.net/fibonacci//">Fibonacci and Lucas Factorizations</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>
%F a(n) = Sum{d|n and n/d odd} A086600(d) + 1 if 6|n, a Mobius-like transform
%t Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; Table[Length[FactorInteger[Lucas[n]]], {n, 150}]
%o (PARI) a(n)=omega(fibonacci(n-1)+fibonacci(n+1)) \\ _Charles R Greathouse IV_, Sep 14 2015
%o (Magma) [#PrimeDivisors(Lucas(n)): n in [1..100]]; // _Vincenzo Librandi_, Jul 26 2017
%Y Cf. A000204 (Lucas numbers), A086599 (number of prime factors, counting multiplicity), A086600 (number of primitive prime factors).
%Y Cf. A053027, A053028, A053032.
%K hard,nonn
%O 1,6
%A _T. D. Noe_, Jul 24 2003
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