Search: id:A086597
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%I A086597
%S A086597 0,0,1,1,1,0,1,1,1,1,1,0,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,2,1,1,1,1,
%T A086597 1,3,1,1,1,2,1,1,2,1,2,1,1,2,2,1,1,2,1,2,1,2,2,2,1,2,1,1,2,1,1,3,2,3,2,
%U A086597 2,1,2,1,1,1,2,2,2,2,3,1,1,2,2,2,2,3,2,2,2,2,1,1,3,2,4,1,2,2,2,2,3,2,1
%N A086597 Number of primitive prime factors in Fibonacci(n).
%C A086597 A prime factor of Fibonacci(n) is called primitive if it does not divide 
               Fibonacci(r) for any r < n. It can be shown that there is at least 
               one primitive prime factor for n > 12. When n is prime, all the prime 
               factors of Fibonacci(n) are primitive; see A080345 for a count of 
               these.
%H A086597 T. D. Noe, <a href="b086597.txt">Table of n, a(n) for n=1..1000</a> (using 
               Blair Kelly's data)
%H A086597 Blair Kelly, <a href="http://home.att.net/~blair.kelly/mathematics/fibonacci/
               ">Fibonacci and Lucas Factorizations</a>
%H A086597 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FibonacciNumber.html">Fibonacci Number</a>
%F A086597 a(n) = Sum{d|n} mu(n/d) A022307(d), inverse Mobius transform of A022307.
%e A086597 a(19) = 2 because Fibonacci(19) = 37*113 and neither 37 nor 113 divide 
               a smaller Fibonacci number.
%t A086597 pLst={}; Join[{0, 0}, Table[f=Transpose[FactorInteger[Fibonacci[n]]][[1]]; 
               f=Complement[f, pLst]; cnt=Length[f]; pLst=Union[pLst, f]; cnt, {n, 
               3, 150}]]
%Y A086597 Cf. A022307 (number of distinct prime factors), A038575 (number of prime 
               factors, counting multiplicity), A061446 (primitive part of Fibonacci(n)), 
               A080345.
%Y A086597 Sequence in context: A053150 A163379 A006466 this_sequence A031214 A056059 
               A158819
%Y A086597 Adjacent sequences: A086594 A086595 A086596 this_sequence A086598 A086599 
               A086600
%K A086597 nonn
%O A086597 1,19
%A A086597 T. D. Noe (noe(AT)sspectra.com), Jul 24 2003

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