login
This site is supported by donations to The OEIS Foundation.

 

Logo

"Email this user" was broken Aug 14 to 9am Aug 16. If you sent someone a message in this period, please send it again.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A022307 Number of distinct prime factors of n-th Fibonacci number. 21
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 (list; graph; refs; listen; history; text; internal format)
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 [Other than 0, this is sequence A060320. - Jon E. Schoenfield, Dec 30 2016]

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 1-8.

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), 449-461.

J. C. Lagarias, Errata to: The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 162, No. 2, (1994), 393-396.

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. 267-284.

Eric Weisstein's World of Mathematics, Fibonacci Number

FORMULA

a(n) = Sum{d|n} 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

(MAGMA) [0] cat [#PrimeDivisors(Fibonacci(n)): n in [1..100]]; // Vincenzo Librandi, Jul 26 2017

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)).

Cf. A060320. - Jon E. Schoenfield, Dec 30 2016

Sequence in context: A238529 A024935 A195150 * A029413 A237523 A238568

Adjacent sequences:  A022304 A022305 A022306 * A022308 A022309 A022310

KEYWORD

nonn

AUTHOR

Clark Kimberling

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified August 17 23:58 EDT 2017. Contains 290682 sequences.