A272948 Positions of Fibonacci numbers in ordered sequence A160009 of all products of Fibonacci numbers.
1, 2, 3, 4, 5, 7, 9, 12, 16, 21, 27, 35, 44, 56, 70, 87, 108, 133, 163, 199, 242, 292, 352, 421, 504, 599, 712, 841, 994, 1167, 1371, 1602, 1873, 2179, 2535, 2936, 3401, 3924, 4528, 5206, 5985, 6858, 7857, 8976, 10252, 11679, 13299, 15109, 17159, 19446, 22028
Offset: 1
Keywords
Examples
A160009 = (0,1,2,3,5,6,8,10,13,15,16,21,...), so that a = (1,2,3,4,5,7,9,12,...).
Links
- Rémy Sigrist, Table of n, a(n) for n = 1..121
- Distinctness of Fibonacci products
- Rémy Sigrist, PARI program for A272948
Programs
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Mathematica
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 *) isFibonacciQ[n_] := Apply[Or, Map[IntegerQ, Sqrt[{# + 4, # - 4} &[5 n^2]]]]; 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}]] Map[Length, ans] (* A272947 *) Flatten[Position[Map[Length, ans], 1]] (* A272948 *) Map[Apply[Times, #] &, Select[ans, Length[#] == 1 &]] (* A000045 *) Map[Apply[Times, #] &, Select[ans, Length[#] == 2 &]] (* A271354 *) Map[Apply[Times, #] &, Select[ans, Length[#] == 3 &]] (* A272949 *) Map[Apply[Times, #] &, Select[ans, Length[#] == 4 &]] (* A272950 *) (* Peter J. C. Moses, May 11 2016 *)
Extensions
More terms from Rémy Sigrist, Mar 17 2019