A062052 Numbers with 2 odd integers in their Collatz (or 3x+1) trajectory.

5, 10, 20, 21, 40, 42, 80, 84, 85, 160, 168, 170, 320, 336, 340, 341, 640, 672, 680, 682, 1280, 1344, 1360, 1364, 1365, 2560, 2688, 2720, 2728, 2730, 5120, 5376, 5440, 5456, 5460, 5461, 10240, 10752, 10880, 10912, 10920, 10922, 20480, 21504, 21760, 21824

Offset

1,1

Comments

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

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

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

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

Links

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Reinhard Zumkeller)

R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.

J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

Index entries for sequences related to 3x+1 (or Collatz) problem

Index entries for 2-automatic sequences.

Formula

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

Example

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

Mathematica

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 *)

Prog

(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, ", "); ); ))
(Haskell)
import Data.List (elemIndices)
a062052 n = a062052_list !! (n-1)
a062052_list = map (+ 1) $ elemIndices 2 a078719_list
-- Reinhard Zumkeller, Oct 08 2011
(Python)
def a(n):
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        if n not in l:
            l.append(n)
            if n<2: break
        else: break
    return len([i for i in l if i % 2])
print([n for n in range(1, 22001) if a(n)==2]) # Indranil Ghosh, Apr 14 2017

Crossrefs

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

Is this a subset of A115774?

Column k=2 of A354236.

Sequence in context: A300019 A115825 A115774 * A245387 A115799 A072703

Adjacent sequences: A062049 A062050 A062051 * A062053 A062054 A062055

Keyword

nonn

Author

David W. Wilson

Status

approved