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%I #12 Feb 28 2018 15:22:28
%S 0,1,1,2,3,5,8,13,21,34,55,89,4,93,97,190,287,477,764,1241,2005,541,
%T 2546,3087,5633,8720,14353,23073,37426,60499,97925,26404,124329,
%U 150733,275062,425795,700857,1126652,1827509,2954161,796945,3751106,4548051,8299157,12847208,21146365,33993573
%N 6-free Fibonacci numbers.
%C The sequences of n-free Fibonacci numbers were suggested by John H. Conway.
%C a(n) is the sum of the two previous terms divided by the largest possible power of 6.
%C 4-free Fibonacci numbers are A224382.
%C The sequence coincides with the Fibonacci sequence until the first multiple of 6 in the Fibonacci sequence: 144, which in this sequence is divided by 36 to produce 4.
%C 7-free Fibonacci numbers is A078414.
%H B. Avila, T. Khovanova, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">Free Fibonacci Sequences</a>, J. Int. Seq. 17 (2014) # 14.8.5.
%t sixPower[n_] := (a = Transpose[FactorInteger[n]]; a2 = Position[a[[1]], 2]; a3 = Position[a[[1]], 3]; If[Length[a2] == 0 || Length[a3] == 0 , res = 0, res = Min[a[[2]][[a2[[1]][[1]]]], a[[2]][[a3[[1]][[1]]]]]]; res); sixFree[n_] := n/6^sixPower[n]; appendNext6Free[list_] := Append[list, sixFree[list[[-1]] + list[[-2]]]]; Nest[appendNext6Free, {0, 1}, 50]
%Y Cf. A224382, A214684.
%K nonn
%O 0,4
%A _Brandon Avila_ and _Tanya Khovanova_, Nov 27 2013
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