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%I #9 Jul 03 2018 21:00:08
%S 1,2,3,4,5,7,8,8,11,10,12,14,14,16,16,16,18,22,22,21,21,25,25,28,29,
%T 28,32,33,33,33,36,32,39,37,37,44,44,44,47,42,48,42,53,50,50,50,54,56,
%U 59,59,59,56,56,64,64,67,71,67,71,67,67,72,72,64,79,79,79,75
%N a(n) = [2^A006666(n)/3^A006667(n)], where [x] = floor(x).
%C The sequence contains A211981 and the powers of 2 (A000079).
%C There exists a subset E = { 1, 2, 3, 4, 5, 8, 10, 16, 21, 32, 42, 64, 85, 128, 170, 227, 256, 341, 512, 682, 1024, 2048, ...} in {a(n)} such that each element m of E generates the Collatz sequence of iterates m -> T_1(m) -> T_2(m) -> T_3(m) -> ... -> 1 where any T_i(m) is an element of E of the form [2^i /3^j] where i = A006666(m), or A006666(m)-1, or ... and j = A006667(m), or A006667(m)-1, or ..., but with A006667(m) <= 3. If m is even then m/2 is in E.
%C For example, the statement that "3 is an element of E" implies that each element of the trajectory 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 belongs to E. Thus the trajectory of the number 3 can be represented by [2^5/3^2] -> [2^5/3^1] -> [2^4/3^1] -> [2^4/3^0] -> [2^3/3^0] -> [2^2/3^0] -> [2^1/3^0] -> [2^0/3^0].
%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e a(39) = [2^A006666(39)/3^A006667(39)] = [2^23/3^11] = [47.353937...] = 47.
%t Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 70; t = {}; n = 0; While[Length[t] < nn, n++; c = Collatz[n]; ev = Length[Select[c, EvenQ]]; od = Length[c] - ev - 1; AppendTo[t, Floor[2^ev/3^od]]]; t
%o (PARI) a(n) = my(t, h); while(n>1, if(n%2, n=3*n+1; t++, n>>=1; h++)); 2^h\3^t; \\ _Michel Marcus_, May 05 2018
%Y Cf. A000079, A006666, A006667, A211981, A225089, A265099.
%K nonn,easy
%O 1,2
%A _Michel Lagneau_, Apr 03 2018
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