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

%I A152012 #25 May 07 2025 13:45:15
%S A152012 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,
%T A152012 33,34,35,36,38,39,40,42,43,45,47,48,51,52,54,56,60,62,63,65,66,72,74,
%U A152012 75,76,82,83,93,94,98,105,106,108,111,112,119,121,122,123,124,125,131
%N A152012 Indices of Fibonacci numbers having exactly one primitive prime factor.
%C A152012 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.
%C A152012 Numbers k such that A086597(k) = 1.
%C A152012 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.
%H A152012 T. D. Noe, <a href="/A152012/b152012.txt">Table of n, a(n) for n=1..392</a>
%t A152012 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 *)
%o A152012 (PARI) isok(pf, vp) = sum(i=1, #pf, vecsearch(vp, pf[i]) == 0) == 1;
%o A152012 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
%Y A152012 Cf. A000045, A001578, A152013.
%K A152012 nonn
%O A152012 1,1
%A A152012 _Max Alekseyev_, Nov 19 2008