

A022307


Number of distinct prime factors of nth Fibonacci number.


18



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, 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, 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
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OFFSET

0,9


COMMENTS

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
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
Row lengths of table A060442.  Reinhard Zumkeller, Aug 30 2014
If k properly divides n then a(n) >= a(k) + 1, except for a(6) = a(3) = 1.  Robert Israel, Aug 18 2015


REFERENCES

Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 18.


LINKS

T. D. Noe, Table of n, a(n) for n = 0..1000 (derived from Kelly's data)
Blair Kelly, Fibonacci and Lucas Factorizations
J. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118. No. 2, (1985), 449461.
J. C. Lagarias, Errata to: The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 162, No. 2, (1994), 393396.
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 1 to 100, n = 101 to 200, n = 201 to 300, n = 301 to 400, n = 401 to 480).
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267284.
Eric Weisstein's World of Mathematics, Fibonacci Number


FORMULA

a(n) = Sum{dn} A086597(d), Mobius transform of A086597.
a(n) = A001221(A000045(n)) = omega(F(n).  Jonathan Vos Post, Dec 06 2006


MATHEMATICA

Table[Length[FactorInteger[Fibonacci[n]]], {n, 150}]


PROG

(PARI) a(n)=omega(fibonacci(n)) \\ Charles R Greathouse IV, Feb 03 2014
(Haskell)
a022307 n = if n == 0 then 0 else a001221 $ a000045 n
 Reinhard Zumkeller, Aug 30 2014


CROSSREFS

Cf. A038575 (number of prime factors, counting multiplicity), A086597 (number of primitive prime factors).
Cf. A000032, A000040, A000045, A001221, A053028.
Cf. A060442, A086598 (omega(Lucas(n)).
Sequence in context: A238529 A024935 A195150 * A029413 A237523 A238568
Adjacent sequences: A022304 A022305 A022306 * A022308 A022309 A022310


KEYWORD

nonn


AUTHOR

Clark Kimberling


STATUS

approved



