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A062052 Numbers with 2 odd integers in their Collatz (or 3x+1) trajectory. 14
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 (list; graph; refs; listen; history; text; internal format)
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

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

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.

Wikipedia, Collatz conjecture

Eric Weisstein's World of Mathematics, Collatz Problem

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

Index entries for 2-automatic sequences.

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 not n in l:

            l+=[n, ]

            if n<2: break

        else: break

    return len(list(filter(lambda i: i%2==1, l)))

print [n for n in xrange(1, 22001) if a(n)==2] # Indranil Ghosh, Apr 14 2017

CROSSREFS

Cf. A062053-A062060.

Is this a subset of A115774?

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

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Last modified November 14 19:12 EST 2018. Contains 317214 sequences. (Running on oeis4.)