A232666 6-free Fibonacci numbers.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 4, 93, 97, 190, 287, 477, 764, 1241, 2005, 541, 2546, 3087, 5633, 8720, 14353, 23073, 37426, 60499, 97925, 26404, 124329, 150733, 275062, 425795, 700857, 1126652, 1827509, 2954161, 796945, 3751106, 4548051, 8299157, 12847208, 21146365, 33993573

Offset

0,4

Comments

The sequences of n-free Fibonacci numbers were suggested by John H. Conway.

a(n) is the sum of the two previous terms divided by the largest possible power of 6.

4-free Fibonacci numbers are A224382.

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.

7-free Fibonacci numbers is A078414.

Links

Table of n, a(n) for n=0..46.

B. Avila, T. Khovanova, Free Fibonacci Sequences, J. Int. Seq. 17 (2014) # 14.8.5.

Mathematica

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]

Crossrefs

Cf. A224382, A214684.

Sequence in context: A132634 A096275 A093089 * A093091 A105471 A189722

Adjacent sequences: A232663 A232664 A232665 * A232667 A232668 A232669

Keyword

nonn

Author

Brandon Avila and Tanya Khovanova, Nov 27 2013

Status

approved