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Search: id:A086597
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| A086597 |
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Number of primitive prime factors in Fibonacci(n). |
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+0
7
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| 0, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 1, 2, 1, 1, 3, 2, 3, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 3, 1, 1, 2, 2, 2, 2, 3, 2, 2, 2, 2, 1, 1, 3, 2, 4, 1, 2, 2, 2, 2, 3, 2, 1
(list; graph; listen)
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OFFSET
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1,19
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COMMENT
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A prime factor of Fibonacci(n) is called primitive if it does not divide Fibonacci(r) for any r < n. It can be shown that there is at least one primitive prime factor for n > 12. When n is prime, all the prime factors of Fibonacci(n) are primitive; see A080345 for a count of these.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000 (using Blair Kelly's data)
Blair Kelly, Fibonacci and Lucas Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number
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FORMULA
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a(n) = Sum{d|n} mu(n/d) A022307(d), inverse Mobius transform of A022307.
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EXAMPLE
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a(19) = 2 because Fibonacci(19) = 37*113 and neither 37 nor 113 divide a smaller Fibonacci number.
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MATHEMATICA
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pLst={}; Join[{0, 0}, Table[f=Transpose[FactorInteger[Fibonacci[n]]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 3, 150}]]
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CROSSREFS
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Cf. A022307 (number of distinct prime factors), A038575 (number of prime factors, counting multiplicity), A061446 (primitive part of Fibonacci(n)), A080345.
Sequence in context: A053150 A163379 A006466 this_sequence A031214 A056059 A158819
Adjacent sequences: A086594 A086595 A086596 this_sequence A086598 A086599 A086600
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KEYWORD
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nonn
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AUTHOR
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T. D. Noe (noe(AT)sspectra.com), Jul 24 2003
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