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

%I A262341 #15 Jul 07 2016 23:45:06
%S A262341 1,1,2,3,5,1,13,7,17,11,89,1,233,29,61,47,1597,19,113,41,421,199,
%T A262341 28657,23,3001,521,109,281,514229,31,2417,2207,19801,3571,141961,107,
%U A262341 2221,9349,135721,2161,59369,211,433494437,307,109441,461,2971215073,1103,6168709,151
%N A262341 Largest primitive prime factor of Fibonacci number F(n), or 1 if no primitive.
%C A262341 Carmichael proved that a(n) > 1 if n > 12.
%C A262341 See A001578 (smallest primitive prime factor of F(n)) and A061446 (primitive part of F(n)) for more links.
%H A262341 R. D. Carmichael, <a href="https://www.jstor.org/stable/1967797?seq=1#page_scan_tab_contents">On the numerical factors of the arithmetic forms α^n±β^n</a>, Annals of Math., 15 (1913), 30-70.
%H A262341 Blair Kelly, <a href="http://mersennus.net/fibonacci//">Fibonacci and Lucas Factorizations</a>
%H A262341 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibmaths.html#section5.1">Fibonacci numbers and special prime factors</a>
%H A262341 Wikipedia, <a href="https://en.wikipedia.org/wiki/Carmichael%27s_theorem">Carmichael's theorem</a>
%e A262341 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.
%t A262341 prms={}; Table[f=First/@FactorInteger[Fibonacci[n]]; p=Complement[f, prms]; prms=Join[prms, p]; If[p=={}, 1, Last[p]], {n, 50}]
%o A262341 (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
%Y A262341 Cf. A000045, A001578, A061446, A086597, A152012, A152013.
%K A262341 nonn
%O A262341 1,3
%A A262341 _Jonathan Sondow_, Oct 12 2015