A262341 Largest primitive prime factor of Fibonacci number F(n), or 1 if no primitive.

1, 1, 2, 3, 5, 1, 13, 7, 17, 11, 89, 1, 233, 29, 61, 47, 1597, 19, 113, 41, 421, 199, 28657, 23, 3001, 521, 109, 281, 514229, 31, 2417, 2207, 19801, 3571, 141961, 107, 2221, 9349, 135721, 2161, 59369, 211, 433494437, 307, 109441, 461, 2971215073, 1103, 6168709, 151

Offset: 1

Author

Jonathan Sondow, Oct 12 2015

Keywords

• nonn

Comments

Carmichael proved that a(n) > 1 if n > 12.

See A001578 (smallest primitive prime factor of F(n)) and A061446 (primitive part of F(n)) for more links.

Examples

			The prime factors of F(46)= 139 * 461 * 28657 that do not divide any smaller Fibonacci number are 139 and 461, so a(46) = 461.
		

Links

• R. D. Carmichael, On the numerical factors of the arithmetic forms α^n±β^n, Annals of Math., 15 (1913), 30-70.
• Blair Kelly, Fibonacci and Lucas Factorizations
• Ron Knott, Fibonacci numbers and special prime factors
• Wikipedia, Carmichael's theorem

Crossrefs

Cf. A000045, A001578, A061446, A086597, A152012, A152013.

Programs

Mathematica

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

Perl

use ntheory ":all"; my %s; for (1..100) { my @f = factor(lucasu(1,-1,$)); pop @f while @f && $s{$f[-1]}++; say "$ ", $f[-1] || 1; }  # Dana Jacobsen, Oct 13 2015