A086597 Number of primitive prime factors in Fibonacci(n).

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

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 A369008 A238015

Adjacent sequences: A086594 A086595 A086596 * A086598 A086599 A086600

Keyword

nonn

Author

T. D. Noe, Jul 24 2003

Status

approved