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A233526 Start with a(1) = 1, a(2) = 3, then a(n)*2^k = a(n+1) + a(n+2), with 2^k the smallest power of 2 (k>0) such that all terms a(n) are positive integers. 3

%I

%S 1,3,1,5,3,7,5,9,1,17,15,19,11,27,17,37,31,43,19,67,9,125,19,231,73,

%T 389,195,583,197,969,607,1331,1097,1565,629,2501

%N Start with a(1) = 1, a(2) = 3, then a(n)*2^k = a(n+1) + a(n+2), with 2^k the smallest power of 2 (k>0) such that all terms a(n) are positive integers.

%C Define 2-free Fibonacci numbers as sequences where b(n) = (b(n-1) + b(n-2))/2^i such that 2^i is the greatest power of 2 that divides b(n-1) + b(n-2). Read backwards from the n-th term, this sequence produces a subsequence of 2-free Fibonacci numbers where we must divide by a power of 2 every every time we add.

%C For other examples of n-free Fibonacci numbers, see A232666, A214684, A224382.

%H Brandon Avila, <a href="/A233526/b233526.txt">Table of n, a(n) for n = 1..1000</a>

%H B. Avila and T. Khovanova, <a href="http://arxiv.org/abs/1403.4614">Free Fibonacci Sequences</a>, arXiv preprint arXiv:1403.4614 [math.NT], 2014 and <a href="https://cs.uwaterloo.ca/journals/JIS/VOL17/Avila/avila4.html">J. Int. Seq. 17 (2014) # 14.8.5</a>.

%o (Python)

%o def minDivisionRich(n, a=1, b=3):

%o ....yield a

%o ....yield b

%o ....for i in range(2, n):

%o ........a *= 2

%o ........while a <= b:

%o ............a *= 2

%o ........a, b = b, a - b

%o ........yield b

%Y Cf. A233525.

%K nonn

%O 1,2

%A _Brandon Avila_, _Tanya Khovanova_, Dec 11 2013

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Last modified May 26 18:08 EDT 2020. Contains 334630 sequences. (Running on oeis4.)