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A038575 Number of prime factors of n-th Fibonacci number, counted with multiplicity.

%I A038575 #37 Feb 16 2025 08:32:38
%S A038575 0,0,0,1,1,1,3,1,2,2,2,1,6,1,2,3,3,1,5,2,4,3,2,1,9,3,2,4,4,1,7,2,4,3,
%T A038575 2,3,10,3,3,3,6,2,7,1,5,5,3,1,12,3,6,3,4,2,8,4,7,5,3,2,12,2,3,5,6,3,7,
%U A038575 3,5,5,7,2,14,2,4,6,5,4,8,2,9,7,3,1,13,4,3,4,9,2,12,5,6,4,2,6,16,4,5,6,10,2,8
%N A038575 Number of prime factors of n-th Fibonacci number, counted with multiplicity.
%C A038575 Row lengths of table A060441. - _Reinhard Zumkeller_, Aug 30 2014
%H A038575 Amiram Eldar, <a href="/A038575/b038575.txt">Table of n, a(n) for n = 0..1408</a> (terms 0..1000 from T. D. Noe derived from Kelly's data)
%H A038575 Blair Kelly, <a href="http://mersennus.net/fibonacci/">Fibonacci and Lucas Factorizations</a>.
%H A038575 Douglas Lind, <a href="https://fq.math.ca/Scanned/6-6/advanced6-6.pdf">Problem H-145</a>, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 6, No. 6 (1968), p. 351; <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/8-4/advanced8-4.pdf">Factor Analysis</a>, Solution to Problem H-145 by the proposer, ibid., Vol. 8, No. 4 (1970), pp. 386-387.
%H A038575 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>.
%F A038575 For n > 0: a(n) = A001222(A000045(n)). - _Reinhard Zumkeller_, Aug 30 2014
%F A038575 a(n) >= A001222(n) - 1 (Lind, 1968). - _Amiram Eldar_, Feb 02 2022
%e A038575 a(12) = 6 because Fibonacci(12) = 144 = 2^4 * 3^2 has 6 prime factors.
%p A038575 with(numtheory):with(combinat):a:=proc(n) if n=0 then 0 else bigomega(fibonacci(n)) fi end: seq(a(n), n=0..102); # _Zerinvary Lajos_, Apr 11 2008
%t A038575 Join[{0, 0}, Table[Plus@@(Transpose[FactorInteger[Fibonacci[n]]][[2]]), {n, 3, 102}]]
%t A038575 Join[{0},PrimeOmega[Fibonacci[Range[110]]]] (* _Harvey P. Dale_, Apr 14 2018 *)
%o A038575 (Haskell)
%o A038575 a038575 n = if n == 0 then 0 else a001222 $ a000045 n
%o A038575 -- _Reinhard Zumkeller_, Aug 30 2014
%o A038575 (PARI) a(n)=bigomega(fibonacci(n)) \\ _Charles R Greathouse IV_, Sep 14 2015
%o A038575 (Python)
%o A038575 from sympy import primeomega, fibonacci
%o A038575 def a(n): return 0 if n == 0 else primeomega(fibonacci(n))
%o A038575 print([a(n) for n in range(103)]) # _Michael S. Branicky_, Feb 02 2022
%Y A038575 Cf. A022307 (number of distinct prime factors), A086597 (number of primitive prime factors).
%Y A038575 Cf. also A001222, A000045, A060441.
%K A038575 nonn
%O A038575 0,7
%A A038575 _Jeff Burch_