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Search: id:A022307
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| A022307 |
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Number of distinct prime factors of n-th Fibonacci number. |
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14
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| 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; listen)
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OFFSET
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0,9
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COMMENT
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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. How many distinct primes divide the n-th Lucas number A000032(n)? - Jonathan Vos Post (jvospost3(AT)gmail.com), 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 (rgwv(AT)rgwv.com), Dec 10 2006
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REFERENCES
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Brother Alfred Brousseau, Fibonacci and Related Number Theoretic Tables, The Fibonacci Association, 1972, pages 1-8.
L. C. Lagarias, The set of primes dividing the Lucas numbers has density 2/3, Pacific J. Math., 118 (1985), 449-461.
Pieter Moree, Counting Divisors of Lucas Numbers, Pacific J. Math, Vol. 186, No. 2, 1998, pp. 267-284.
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LINKS
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T. D. Noe, Table of n, a(n) for n=0..1000 (derived from Kelly's data)
Blair Kelly, Fibonacci and Lucas Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 1 to 100).
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 101 to 200).
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 201 to 300).
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 301 to 400).
Hisanori Mishima, WIFC (World Integer Factorization Center), Fibonacci numbers (n = 401 to 480).
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FORMULA
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a(n) = Sum{d|n} A086597(d), Mobius transform of A086597.
A001221(A000045(n)) = omega(F(n). - Jonathan Vos Post (jvospost3(AT)gmail.com), Dec 06 2006
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MATHEMATICA
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Table[Length[FactorInteger[Fibonacci[n]]], {n, 150}]
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CROSSREFS
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Cf. A038575 (number of prime factors, counting multiplicity), A086597 (number of primitive prime factors).
Cf. A000032, A000040, A000045, A001221, A053028.
Sequence in context: A080354 A024935 A137921 this_sequence A029413 A105154 A076447
Adjacent sequences: A022304 A022305 A022306 this_sequence A022308 A022309 A022310
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KEYWORD
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nonn
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AUTHOR
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Clark Kimberling (ck6(AT)evansville.edu)
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