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A272947 Number of factors Fibonacci(i) > 1 of A160009(n+1).

%I A272947 #11 Feb 17 2018 20:04:12
%S A272947 1,1,1,1,1,2,1,2,1,2,2,1,2,2,3,1,2,2,2,3,1,2,2,2,3,3,1,2,2,2,2,3,3,3,
%T A272947 1,2,2,2,2,3,3,3,3,1,4,2,2,2,2,2,3,3,3,3,3,1,4,2,2,2,2,2,3,3,3,3,3,3,
%U A272947 3,1,4,4,2,2,2,2,2,2,3,3,3,3,3,3,3,3
%N A272947 Number of factors Fibonacci(i) > 1 of A160009(n+1).
%H A272947 MathOverflow, <a href="http://mathoverflow.net/questions/238505/distinctness-of-products-of-fibonacci-numbers">Distinctness of products of Fibonacci numbers</a>
%e A272947 A160009(15) = 30 = 2*3*5, so that a(15) = 3.
%t A272947 s = {1}; nn = 60; f = Fibonacci[2 + Range[nn]]; Do[s = Union[s, Select[s*f[[i]], # <= f[[nn]] &]], {i, nn}]; s =  Prepend[s, 0]; Take[s, 100]  (* A160009 *)
%t A272947 isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
%t A272947 ans = Join[{{0}}, {{1}}, Table[#[[Flatten[Position[Map[Apply[Times, #] &, #], s[[n]]]][[1]]]] &[Rest[Subsets[Rest[Map[#[[1]] &, Select[Map[{#, isFibonacciQ[#]} &, Divisors[s[[n]]]], #[[2]] &]]]]]], {n, 3, 500}]]
%t A272947 Map[Length, ans] (* A272947 *)
%t A272947 Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
%t A272947 Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
%t A272947 Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
%t A272947 Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
%t A272947 Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
%t A272947 (* _Peter J. C. Moses_, May 11 2016 *)
%Y A272947 Cf. A000045, A160009, A272948, A271354, A272949, A273950.
%K A272947 nonn,easy
%O A272947 1,6
%A A272947 _Clark Kimberling_, May 13 2016