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A062053 Numbers with exactly 3 odd integers in their Collatz (or 3x+1) trajectory.

%I A062053 #52 Feb 16 2025 08:32:44
%S A062053 3,6,12,13,24,26,48,52,53,96,104,106,113,192,208,212,213,226,227,384,
%T A062053 416,424,426,452,453,454,768,832,848,852,853,904,906,908,909,1536,
%U A062053 1664,1696,1704,1706,1808,1812,1813,1816,1818,3072,3328,3392,3408,3412,3413,3616
%N A062053 Numbers with exactly 3 odd integers in their Collatz (or 3x+1) trajectory.
%C A062053 The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd (A006370).
%C A062053 The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.
%C A062053 A078719(a(n)) = 3; A006667(a(n)) = 2.
%D A062053 J. R. Goodwin, Results on the Collatz Conjecture, Sci. Ann. Comput. Sci. 13 (2003) pp. 1-16
%D A062053 J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.
%H A062053 Reinhard Zumkeller and David A. Corneth, <a href="/A062053/b062053.txt">Table of n, a(n) for n = 1..16191</a> (first 250 terms from Reinhard Zumkeller, terms < 10^25)
%H A062053 J. Shallit and D. Wilson, <a href="http://www.cs.uwaterloo.ca/~shallit/Papers/wilson.ps">The "3x+1" Problem and Finite Automata</a>, Bulletin of the EATCS #46 (1992) pp. 182-185.
%H A062053 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>
%H A062053 Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>
%H A062053 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%H A062053 <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.
%F A062053 The two formulas giving this sequence are listed in Corollary 3.1 and Corollary 3.2 in J. R. Goodwin with the following caveats: the value x cannot equal zero in Corollary 3.2, one must multiply the formulas by all powers of 2 (2^1, 2^2, ...) to get the evens. - _Jeffrey R. Goodwin_, Oct 26 2011
%e A062053 The Collatz trajectory of 3 is (3,10,5,16,8,4,2,1), which contains 3 odd integers.
%t A062053 Collatz[n_?OddQ] := (3n + 1)/2; Collatz[n_?EvenQ] := n/2; oddIntCollatzCount[n_] := Length[Select[NestWhileList[Collatz, n, # != 1 &], OddQ]]; Select[Range[4000], oddIntCollatzCount[#] == 3 &] (* _Alonso del Arte_, Oct 28 2011 *)
%o A062053 (Haskell)
%o A062053 import Data.List (elemIndices)
%o A062053 a062053 n = a062053_list !! (n-1)
%o A062053 a062053_list = map (+ 1) $ elemIndices 3 a078719_list
%o A062053 -- _Reinhard Zumkeller_, Oct 08 2011
%Y A062053 Cf. A006370, A078719, A006667.
%Y A062053 Cf. A000079, A062052, A062054, A062055, A062056, A062057, A062058, A062059, A062060.
%Y A062053 Cf. A198584 (this sequence without the even numbers).
%Y A062053 See also A198587.
%Y A062053 Column k=3 of A354236.
%K A062053 nonn,easy
%O A062053 1,1
%A A062053 _David W. Wilson_