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 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 (list; graph; refs; listen; history; text; internal format)
 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), 30-70. Blair Kelly, Fibonacci and Lucas Factorizations Eric Weisstein's World of Mathematics, Fibonacci Number FORMULA a(n) = Sum{d|n} 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: A006466 A316439 A335078 * A322320 A238015 A257679 Adjacent sequences:  A086594 A086595 A086596 * A086598 A086599 A086600 KEYWORD nonn AUTHOR T. D. Noe, Jul 24 2003 STATUS approved

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Last modified September 20 14:44 EDT 2021. Contains 347586 sequences. (Running on oeis4.)