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A272948 Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.

%I A272948 #9 Mar 17 2019 12:21:46
%S A272948 1,2,3,4,5,7,9,12,16,21,27,35,44,56,70,87,108,133,163,199,242,292,352,
%T A272948 421,504,599,712,841,994,1167,1371,1602,1873,2179,2535,2936,3401,3924,
%U A272948 4528,5206,5985,6858,7857,8976,10252,11679,13299,15109,17159,19446,22028
%N A272948 Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.
%H A272948 Rémy Sigrist, <a href="/A272948/b272948.txt">Table of n, a(n) for n = 1..121</a>
%H A272948 <a href="http://mathoverflow.net/questions/238505/distinctness-of-products-of-fibonacci-numbers">Distinctness of Fibonacci products</a>
%H A272948 Rémy Sigrist, <a href="/A272948/a272948.gp.txt">PARI program for A272948</a>
%e A272948 A160009 = (0,1,2,3,5,6,8,10,13,15,16,21,...), so that a = (1,2,3,4,5,7,9,12,...).
%t A272948 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 A272948 isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]];
%t A272948 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 A272948 Map[Length, ans] (* A272947 *)
%t A272948 Flatten[Position[Map[Length, ans], 1]]  (* A272948 *)
%t A272948 Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]]  (* A000045 *)
%t A272948 Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]]  (* A271354 *)
%t A272948 Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]]  (* A272949 *)
%t A272948 Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]]  (* A272950 *)
%t A272948 (* _Peter J. C. Moses_, May 11 2016 *)
%Y A272948 Cf. A000045, A160009, A272947, A271354, A272949, A273950.
%K A272948 nonn
%O A272948 1,2
%A A272948 _Clark Kimberling_, May 13 2016
%E A272948 More terms from _Rémy Sigrist_, Mar 17 2019