A001578 Smallest primitive prime factor of Fibonacci number F(n), or 1 if F(n) has no primitive prime factor.

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 37, 41, 421, 199, 28657, 23, 3001, 521, 53, 281, 514229, 31, 557, 2207, 19801, 3571, 141961, 107, 73, 9349, 135721, 2161, 2789, 211, 433494437, 43, 109441, 139, 2971215073, 1103, 97, 101

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn

Comments

A prime factor of F(n) is called primitive if it does not divide F(r) for any r < n.

A Fibonacci number can have more than one primitive factor; the primitive factors of F(19) are 37 and 113.

From Robert Israel, Oct 13 2015: (Start)

Since gcd(F(n),F(k)) = F(gcd(n,k)), the non-primitive prime factors of F(n) are factors of F(k) for some proper divisors k of n.

Since prime p divides F(p-1) if p == 1 or 4 (mod 5), F(p+1) if p == 2 or 3 mod 5, F(p) if p = 5, we have a(n) >= n-1 if a(n) > 1.

a(n) = n-1 iff n=2 or n-1 is in A000057.

a(n) = n+1 iff n+1 is a prime in A106535. (End)

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Max Alekseyev, Table of n, a(n) for n = 1..1452 (terms 1..1000 from Blair Kelly and T. D. Noe, terms 1409..1422 from Tyler Busby)
• Mansur S. Boase, A Result About the Primes Dividing Fibonacci Numbers, The Fibonacci Quarterly, 39.5 (2001) 386.
• R. D. Carmichael, On the numerical factors of the arithmetic forms α^n ± β^n, Annals of Math., 15 (1/4) (1913), 30-70.
• D. Jarden, On the greatest primitive divisors of Fibonacci and Lucas numbers with prime-power subscripts, Fib. Quart. 1(#3) (1963), 15-31.
• Blair Kelly, Fibonacci and Lucas Factorizations

Crossrefs

Cf. A000045, A000057, A106535, A086597 (number of primitive prime factors in F(n)), A061488 (1's omitted), A262341 (largest primitive prime factor of F(n)).

Programs

Maple

for n from 1 to 350 do
  f:= combinat:-fibonacci(n);
  if not isprime(n) then
    for k in map(t -> n/t, numtheory:-factorset(n)) do
       fk:= combinat:-fibonacci(k);
       g:= igcd(f,fk);
       while g > 1 do
         f:= f/g;
         g:= igcd(f,fk);
       od
    od
  fi;
  if f = 1 then A[n]:= 1; next fi;
  F:= map(t -> t[1],ifactors(f,easy)[2]);
  p:= select(type, F,integer);
  if nops(p) >= 1 then A[n]:= min(p); next fi;
  A[n]:= min(numtheory:-factorset(f));
od:
seq(A[i],i=1..350); # Robert Israel, Oct 13 2015

Mathematica

prms={}; Table[f=First/@FactorInteger[Fibonacci[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, First[p]], {n, 50}]

PARI

a(n) = {my(v = vector(n, k, fibonacci(k))); my(vf = vector(n, k, factor(v[k])[,1]~)); for (k=1, n-1, vf[n] = setminus(vf[n], vf[k]);); if (#vf[n], vecmin(vf[n]), 1);} \\ Michel Marcus, May 11 2021

Formula

a(n) = 1 if and only if n = 1, 2, 6, or 12, by Carmichael's theorem. - Jonathan Sondow, Dec 07 2017

Extensions

Edited by T. D. Noe, Apr 15 2004

Definition clarified at the suggestion of Joerg Arndt by Jonathan Sondow, Oct 13 2015