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A086600 Number of primitive prime factors in the n-th Lucas number A000204(n).

%I A086600 #35 Jun 15 2025 10:57:19
%S A086600 0,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,2,1,1,1,2,1,1,2,1,2,
%T A086600 2,1,1,1,2,2,1,2,2,3,2,2,1,2,1,2,2,2,1,1,2,1,2,2,3,2,1,1,2,2,3,1,2,1,
%U A086600 2,1,1,1,2,2,2,1,2,1,1,1,2,3,2,2,2,1,2,2,2,1,1,2,2,3,3,1,2,2,3,2,3,2,3,3,2
%N A086600 Number of primitive prime factors in the n-th Lucas number A000204(n).
%C A086600 A prime factor of Lucas(n) is called primitive if it does not divide Lucas(r) for any r < n. It can be shown that there is at least one primitive prime factor for n > 6. When n is prime, all the prime factors of Lucas(n) are primitive.
%H A086600 Max Alekseyev, <a href="/A086600/b086600.txt">Table of n, a(n) for n = 1..1411</a> (first 1000 terms from T. D. Noe)
%H A086600 Blair Kelly, <a href="http://mersennus.net/fibonacci//">Fibonacci and Lucas Factorizations</a>,
%H A086600 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LucasNumber.html">Lucas Number</a>.
%F A086600 a(n) = Sum{d|n and n/d odd} mu(n/d) a(d) -1 if 6|n and n/6 is a power of 2.
%e A086600 a(22) = 2 because Lucas(22) = 3*43*307 and neither 43 nor 307 divide a smaller Lucas number.
%t A086600 Lucas[n_] := Fibonacci[n+1] + Fibonacci[n-1]; pLst={}; Join[{0}, Table[f=Transpose[FactorInteger[Lucas[n]]][[1]]; f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 2, 150}]]
%o A086600 (Magma) lst:=[]; pr:=1; for n in [1..105] do pd:=PrimeDivisors(Lucas(n)); d:=1; t:=0; for c in [1..#pd] do f:=pd[c]; if Gcd(pr, f) eq 1 then t+:=1; else d:=d*f; end if; end for; Append(~lst, t); pr:=pr*Truncate(Lucas(n)/d); end for; lst; // _Arkadiusz Wesolowski_, Jun 22 2016
%Y A086600 Cf. A000204 (Lucas numbers), A058036, A086598 (number of distinct prime factors), A086599 (number of prime factors, counting multiplicity), A274333.
%K A086600 hard,nonn
%O A086600 1,22
%A A086600 _T. D. Noe_, Jul 24 2003