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

Author

T. D. Noe, Jul 24 2003

Keywords

• nonn

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.

Examples

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

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

Crossrefs

Cf. A022307 (number of distinct prime factors), A038575 (number of prime factors, counting multiplicity), A061446 (primitive part of Fibonacci(n)), A080345.

Programs

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}]]

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

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