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URL: https://oeis.org/A001578

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A001578
Smallest primitive prime factor of Fibonacci number F(n), or 1 if F(n) has no primitive prime factor.
14
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
1,3
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.
FORMULA
a(n) = 1 if and only if n = 1, 2, 6, or 12, by Carmichael's theorem. - Jonathan Sondow, Dec 07 2017
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}]
PROG
(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
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)).
Sequence in context: A276350 A262215 A030790 * A262341 A178763 A111141
KEYWORD
nonn
EXTENSIONS
Edited by T. D. Noe, Apr 15 2004
Definition clarified at the suggestion of Joerg Arndt by Jonathan Sondow, Oct 13 2015
STATUS
approved