A152013 -id:A152013 - cp's OEIS frontend

A152012 Indices of Fibonacci numbers having exactly one primitive prime factor.

3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 43, 45, 47, 48, 51, 52, 54, 56, 60, 62, 63, 65, 66, 72, 74, 75, 76, 82, 83, 93, 94, 98, 105, 106, 108, 111, 112, 119, 121, 122, 123, 124, 125, 131

Offset: 1

Max Alekseyev, Nov 19 2008

nonn

It is known that Fibonacci number A000045(n) has a primitive prime factor for all n, except n=0, 1, 2, 6 and 12. This sequence lists such indices n that A000045(n) has exactly one primitive prime factor (equal A001578(n)). Sister sequence A152013 provides indices of Fibonacci numbers with at least 2 prime factors. The current sequence A152012 and its sister sequence A152013 along with the finite set {0,1,2,6,12} form a partition of the natural numbers.
Numbers k such that A086597(k) = 1.
For prime p, all prime factors of Fibonacci(p) are primitive. Hence, the only primes in this sequence are the prime numbers in A001605, which gives the indices of prime Fibonacci numbers.

T. D. Noe, Table of n, a(n) for n=1..392

Cf. A000045, A001578, A152013.

primitivePrimeFactors[n_] := Cases[FactorInteger[Fibonacci[n]][[All, 1]], p_ /; And @@ (GCD[p, #] == 1 & /@ Array[Fibonacci, n-1])]; Reap[For[n=3, n <= 200, n++, If[Length[primitivePrimeFactors[n]] == 1, Print[n]; Sow[n]]]][[2, 1]] (* Jean-François Alcover, Dec 12 2014 *)

isok(pf, vp) = sum(i=1, #pf, vecsearch(vp, pf[i]) == 0) == 1;
lista(nn) = {vp = []; for (n=3, nn, pf = factor(fibonacci(n))[,1]; if (isok(pf, vp), print1(n, ", ")); vp = vecsort(concat(vp, pf),, 8););} \\ Michel Marcus, Nov 29 2014

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

Original entry on oeis.org

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

Jonathan Sondow, Oct 12 2015

nonn

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.

			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.

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

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

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

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

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A152012 Indices of Fibonacci numbers having exactly one primitive prime factor.

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