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Numbers with 2 odd integers in their Collatz (or 3x+1) trajectory.
15

%I #62 May 24 2022 17:50:18

%S 5,10,20,21,40,42,80,84,85,160,168,170,320,336,340,341,640,672,680,

%T 682,1280,1344,1360,1364,1365,2560,2688,2720,2728,2730,5120,5376,5440,

%U 5456,5460,5461,10240,10752,10880,10912,10920,10922,20480,21504,21760,21824

%N Numbers with 2 odd integers in their Collatz (or 3x+1) trajectory.

%C The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd.

%C The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.

%C The sequence consists of terms of A002450 and their 2^k multiples. The first odd integer in the trajectory is one of the terms of A002450 and the second odd one is the terminal 1. - _Antti Karttunen_, Feb 21 2006

%C This sequence looks to appear first in the literature on page 1285 in R. E. Crandall.

%H Reinhard Zumkeller and T. D. Noe, <a href="/A062052/b062052.txt">Table of n, a(n) for n = 1..1000</a> (first 100 terms from Reinhard Zumkeller)

%H R. E. Crandall, <a href="http://dx.doi.org/10.1090/S0025-5718-1978-0480321-3">On the 3x+1 problem</a>, Math. Comp., 32 (1978) 1281-1292.

%H 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/CollatzProblem.html">Collatz Problem</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">Collatz conjecture</a>

%H <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>

%H <a href="/index/Ar#2-automatic">Index entries for 2-automatic sequences</a>.

%F A078719(a(n)) = 2; A006667(a(n)) = 1.

%e The Collatz trajectory of 5 is (5,16,8,4,2,1), which contains 2 odd integers.

%t Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[22000], countOdd[Collatz[#]] == 2 &] (* _T. D. Noe_, Dec 03 2012 *)

%o (PARI) for(n=2,100000,s=n; t=0; while(s!=1,if(s%2==0,s=s/2,s=3*s+1; t++); if(s*t==1,print1(n,","); ); ))

%o (Haskell)

%o import Data.List (elemIndices)

%o a062052 n = a062052_list !! (n-1)

%o a062052_list = map (+ 1) $ elemIndices 2 a078719_list

%o -- _Reinhard Zumkeller_, Oct 08 2011

%o (Python)

%o def a(n):

%o l=[n, ]

%o while True:

%o if n%2==0: n//=2

%o else: n = 3*n + 1

%o if n not in l:

%o l.append(n)

%o if n<2: break

%o else: break

%o return len([i for i in l if i % 2])

%o print([n for n in range(1, 22001) if a(n)==2]) # _Indranil Ghosh_, Apr 14 2017

%Y Cf. A062053, A062054, A062055, A062056, A062057, A062058, A062059, A062060.

%Y Is this a subset of A115774?

%Y Column k=2 of A354236.

%K nonn

%O 1,1

%A _David W. Wilson_