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) = 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
KEYWORD
nonn
AUTHOR
T. D. Noe, Jul 24 2003
STATUS
approved