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

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, 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, 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

Offset: 0

Author

Jeff Burch

Keywords

• nonn

Comments

Row lengths of table A060441. - Reinhard Zumkeller, Aug 30 2014

Examples

			a(12) = 6 because Fibonacci(12) = 144 = 2^4 * 3^2 has 6 prime factors.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1408 (terms 0..1000 from T. D. Noe derived from Kelly's data)
• Blair Kelly, Fibonacci and Lucas Factorizations.
• Douglas Lind, Problem H-145, Advanced Problems and Solutions, The Fibonacci Quarterly, Vol. 6, No. 6 (1968), p. 351; Factor Analysis, Solution to Problem H-145 by the proposer, ibid., Vol. 8, No. 4 (1970), pp. 386-387.
• Eric Weisstein's World of Mathematics, Fibonacci Number.

Crossrefs

Cf. A022307 (number of distinct prime factors), A086597 (number of primitive prime factors).

Cf. also A001222, A000045, A060441.

Programs

Haskell

a038575 n = if n == 0 then 0 else a001222 $ a000045 n
-- Reinhard Zumkeller, Aug 30 2014

Maple

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

Mathematica

Join[{0, 0}, Table[Plus@@(Transpose[FactorInteger[Fibonacci[n]]][[2]]), {n, 3, 102}]]
Join[{0},PrimeOmega[Fibonacci[Range[110]]]] (* Harvey P. Dale, Apr 14 2018 *)

PARI

a(n)=bigomega(fibonacci(n)) \\ Charles R Greathouse IV, Sep 14 2015

Python

from sympy import primeomega, fibonacci
def a(n): return 0 if n == 0 else primeomega(fibonacci(n))
print([a(n) for n in range(103)]) # Michael S. Branicky, Feb 02 2022

Formula

For n > 0: a(n) = A001222(A000045(n)). - Reinhard Zumkeller, Aug 30 2014

a(n) >= A001222(n) - 1 (Lind, 1968). - Amiram Eldar, Feb 02 2022