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

0,7

Comments

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

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.

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

Example

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

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 *)

Prog

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

Crossrefs

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

Cf. also A001222, A000045, A060441.

Sequence in context: A236172 A139381 A225849 * A347430 A346491 A304302

Adjacent sequences: A038572 A038573 A038574 * A038576 A038577 A038578

Keyword

nonn

Author

Jeff Burch

Status

approved