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%I #67 Sep 08 2022 08:44:46
%S 0,0,0,1,1,1,1,1,2,2,2,1,2,1,2,3,3,1,3,2,4,3,2,1,4,2,2,4,4,1,5,2,4,3,
%T 2,3,5,3,3,3,6,2,5,1,5,5,3,1,6,3,5,3,4,2,6,4,6,5,3,2,8,2,3,5,6,3,5,3,
%U 5,5,7,2,8,2,4,5,5,4,6,2,9,7,3,1,9,4,3,4,9,2,10,4,6,4,2,6,9,4,5,6
%N Number of distinct prime factors of n-th Fibonacci number.
%C Although every prime divides some Fibonacci number, this is not true for the Lucas numbers. Exactly 1/3 of all primes do not divide any Lucas number. See Lagarias and Moree for more details. - _Jonathan Vos Post_, Dec 06 2006
%C First occurrence of k: 0, 3, 8, 15, 20, 30, 40, 70, 60, 80, 90, 140, 176, 120, 168, 180, 324, 252, 240, 378, ..., . - _Robert G. Wilson v_, Dec 10 2006 [Other than 0, this is sequence A060320. - _Jon E. Schoenfield_, Dec 30 2016]
%C Row lengths of table A060442. - _Reinhard Zumkeller_, Aug 30 2014
%C If k properly divides n then a(n) >= a(k) + 1, except for a(6) = a(3) = 1. - _Robert Israel_, Aug 18 2015
%D Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.
%H Amiram Eldar, <a href="/A022307/b022307.txt">Table of n, a(n) for n = 0..1408</a> (terms 0..1000 from T. D. Noe derived from Kelly's data)
%H Blair Kelly, <a href="http://mersennus.net/fibonacci//">Fibonacci and Lucas Factorizations</a>
%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102706452">The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 118. No. 2, (1985), 449-461.
%H J. C. Lagarias, <a href="http://projecteuclid.org/euclid.pjm/1102622818">Errata to: The set of primes dividing the Lucas numbers has density 2/3</a>, Pacific J. Math., 162, No. 2, (1994), 393-396.
%H Hisanori Mishima, WIFC (World Integer Factorization Center), <a href="http://www.asahi-net.or.jp/~kc2h-msm/mathland/matha1/matha108.htm">Fibonacci numbers (n = 1 to 100</a>, <a href="http://www.asahi-net.or.jp/~kc2h-msm/mathland/matha1/matha109.htm">n = 101 to 200</a>, <a href="http://www.asahi-net.or.jp/~kc2h-msm/mathland/matha1/matha110.htm">n = 201 to 300</a>, <a href="http://www.asahi-net.or.jp/~kc2h-msm/mathland/matha1/matha111.htm">n = 301 to 400</a>, <a href="http://www.asahi-net.or.jp/~kc2h-msm/mathland/matha1/matha112.htm">n = 401 to 480)</a>.
%H Pieter Moree, <a href="http://dx.doi.org/10.2140/pjm.1998.186.267">Counting Divisors of Lucas Numbers</a>, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%F a(n) = Sum{d|n} A086597(d), Mobius transform of A086597.
%F a(n) = A001221(A000045(n)) = omega(F(n). - _Jonathan Vos Post_, Dec 06 2006
%t Table[Length[FactorInteger[Fibonacci[n]]], {n, 150}]
%o (PARI) a(n)=omega(fibonacci(n)) \\ _Charles R Greathouse IV_, Feb 03 2014
%o (Haskell)
%o a022307 n = if n == 0 then 0 else a001221 $ a000045 n
%o -- _Reinhard Zumkeller_, Aug 30 2014
%o (Magma) [0] cat [#PrimeDivisors(Fibonacci(n)): n in [1..100]]; // _Vincenzo Librandi_, Jul 26 2017
%Y Cf. A038575 (number of prime factors, counting multiplicity), A086597 (number of primitive prime factors).
%Y Cf. A000032, A000040, A000045, A001221, A053028.
%Y Cf. A060442, A086598 (omega(Lucas(n)).
%Y Cf. A060320. - _Jon E. Schoenfield_, Dec 30 2016
%K nonn
%O 0,9
%A _Clark Kimberling_
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