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 A302175 a(n) = [2^A006666(n)/3^A006667(n)], where [x] = floor(x). 0
 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, 28, 32, 33, 33, 33, 36, 32, 39, 37, 37, 44, 44, 44, 47, 42, 48, 42, 53, 50, 50, 50, 54, 56, 59, 59, 59, 56, 56, 64, 64, 67, 71, 67, 71, 67, 67, 72, 72, 64, 79, 79, 79, 75 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS The sequence contains A211981 and the powers of 2 (A000079). 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. 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]. LINKS EXAMPLE a(39) = [2^A006666(39)/3^A006667(39)] = [2^23/3^11] = [47.353937...] = 47. MATHEMATICA 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 PROG (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 CROSSREFS Cf. A000079, A006666, A006667, A211981, A225089, A265099. Sequence in context: A107900 A112922 A305077 * A228683 A133017 A290019 Adjacent sequences:  A302172 A302173 A302174 * A302176 A302177 A302178 KEYWORD nonn,easy AUTHOR Michel Lagneau, Apr 03 2018 STATUS approved

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Last modified April 20 18:08 EDT 2021. Contains 343135 sequences. (Running on oeis4.)