

A086597


Number of primitive prime factors in Fibonacci(n).


14



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
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OFFSET

1,19


COMMENTS

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.


LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000
R. D. Carmichael, On the numerical factors of the arithmetic forms α^n ± β^n, Annals of Math., 15 (1/4) (1913), 3070.
Blair Kelly, Fibonacci and Lucas Factorizations
Eric Weisstein's World of Mathematics, Fibonacci Number


FORMULA

a(n) = Sum{dn} mu(n/d) A022307(d), inverse Mobius transform of A022307.
a(n) = 0 if and only if n = 1, 2, 6, or 12, by Carmichael's theorem.  Jonathan Sondow, Dec 07 2017


EXAMPLE

a(19) = 2 because Fibonacci(19) = 37*113 and neither 37 nor 113 divide a smaller Fibonacci number.


MATHEMATICA

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}]]


PROG

(PARI) a(n)=my(t=fibonacci(n), g); fordiv(n, d, if(d==n, break); g=fibonacci(d); while((g=gcd(g, t))>1, t /= g)); omega(t) \\ Charles R Greathouse IV, Oct 06 2016


CROSSREFS

Cf. A022307 (number of distinct prime factors), A038575 (number of prime factors, counting multiplicity), A061446 (primitive part of Fibonacci(n)), A080345.
Sequence in context: A163379 A006466 A316439 * A322320 A238015 A257679
Adjacent sequences: A086594 A086595 A086596 * A086598 A086599 A086600


KEYWORD

nonn


AUTHOR

T. D. Noe, Jul 24 2003


STATUS

approved



