A006577 Number of halving and tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached. (Formerly M4323)

0, 1, 7, 2, 5, 8, 16, 3, 19, 6, 14, 9, 9, 17, 17, 4, 12, 20, 20, 7, 7, 15, 15, 10, 23, 10, 111, 18, 18, 18, 106, 5, 26, 13, 13, 21, 21, 21, 34, 8, 109, 8, 29, 16, 16, 16, 104, 11, 24, 24, 24, 11, 11, 112, 112, 19, 32, 19, 32, 19, 19, 107, 107, 6, 27, 27, 27, 14, 14, 14, 102, 22

Offset

1,3

Comments

The 3x+1 or Collatz problem is as follows: start with any number n. If n is even, divide it by 2, otherwise multiply it by 3 and add 1. Do we always reach 1? This is a famous unsolved problem. It is conjectured that the answer is yes.

It seems that about half of the terms satisfy a(i) = a(i+1). For example, up to 10000000, 4964705 terms satisfy this condition.

n is an element of row a(n) in triangle A127824. - Reinhard Zumkeller, Oct 03 2012

The number of terms that satisfy a(i) = a(i+1) for i being a power of ten from 10^1 through 10^10 are: 0, 31, 365, 4161, 45022, 477245, 4964705, 51242281, 526051204, 5378743993. - John Mason, Mar 02 2018

5 seems to be the only number whose value matches its total number of steps (checked to n <= 10^9). - Peter Woodward, Feb 15 2021

References

R. K. Guy, Unsolved Problems in Number Theory, E16.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

N. J. A. Sloane, Table of n, a(n) for n = 1..10000

David Eisenbud and Brady Haran, UNCRACKABLE? The Collatz Conjecture, Numberphile video, 2016.

Geometry.net, Links on Collatz Problem

Christian Hercher, There are no Collatz m-Cycles with m <= 91, J. Int. Seq. (2023) Vol. 26, Article 23.3.5.

Jason Holt, Log-log plot of first billion terms

Jason Holt, Plot of 1 billion values of the number of steps to drop below n (A060445), log scale on x axis

Jason Holt, Plot of 10 billion values of the number of steps to drop below n (A060445), log scale on x axis

A. Krowne, Collatz problem, PlanetMath.org.

J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.

J. C. Lagarias, How random are 3x+1 function iterates?, in The Mathemagician and the Pied Puzzler - A Collection in Tribute to Martin Gardner, Ed. E. R. Berlekamp and T. Rogers, A. K. Peters, 1999, pp. 253-266.

J. C. Lagarias, The 3x+1 Problem: an annotated bibliography, II (2000-2009), arXiv:0608208 [math.NT], 2006-2012.

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.

Jeffrey C. Lagarias, The 3x+1 Problem: An Overview, arXiv:2111.02635 [math.NT], 2021.

M. Le Brun, Email to N. J. A. Sloane, Jul 1991

Mathematical BBS, Biblography on Collatz Sequence

P. Picart, Algorithme de Collatz et conjecture de Syracuse

E. Roosendaal, On the 3x+1 problem

J. L. Simons, On the nonexistence of 2-cycles for the 3x+1 problem, Math. Comp. 75 (2005), 1565-1572.

N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 8.

G. Villemin's Almanach of Numbers, Cycle of Syracuse

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz Conjecture

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

Formula

a(n) = A006666(n) + A006667(n).

a(n) = A112695(n) + 2 for n > 2. - Reinhard Zumkeller, Apr 18 2008

a(n) = A008908(n) - 1. - L. Edson Jeffery, Jul 21 2014

a(n) = A135282(n) + A208981(n) (after Alonso del Arte's comment in A208981), if 1 is reached, otherwise a(n) = -1. - Omar E. Pol, Apr 10 2022

Example

a(5)=5 because the trajectory of 5 is (5,16,8,4,2,1).

Maple

A006577 := proc(n)
        local a, traj ;
        a := 0 ;
        traj := n ;
        while traj > 1 do
                if type(traj, 'even') then
                        traj := traj/2 ;
                else
                        traj := 3*traj+1 ;
                end if;
                a := a+1 ;
        end do:
        return a;
end proc: # R. J. Mathar, Jul 08 2012

Mathematica

f[n_] := Module[{a=n, k=0}, While[a!=1, k++; If[EvenQ[a], a=a/2, a=a*3+1]]; k]; Table[f[n], {n, 4!}] (* Vladimir Joseph Stephan Orlovsky, Jan 08 2011 *)
Table[Length[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #!=1&]]-1, {n, 80}] (* Harvey P. Dale, May 21 2012 *)

Prog

(PARI) a(n)=if(n<0, 0, s=n; c=0; while(s>1, s=if(s%2, 3*s+1, s/2); c++); c)
(PARI) step(n)=if(n%2, 3*n+1, n/2);
A006577(n)=if(n==1, 0, A006577(step(n))+1); \\ Michael B. Porter, Jun 05 2010
(Haskell)
import Data.List (findIndex)
import Data.Maybe (fromJust)
a006577 n = fromJust $ findIndex (n `elem`) a127824_tabf
-- Reinhard Zumkeller, Oct 04 2012, Aug 30 2012
(Python)
def a(n):
    if n==1: return 0
    x=0
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        x+=1
        if n<2: break
    return x
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 05 2017
(Python)
def A006577(n):
    ct = 0
    while n != 1: n = A006370(n); ct += 1
    return ct # Ya-Ping Lu, Feb 22 2024

Crossrefs

See A070165 for triangle giving trajectories of n = 1, 2, 3, ....

Cf. A006370, A125731, A127885, A127886, A008908, A112695, A135282, A208981.

See also A008884, A161021, A161022, A161023.

Sequence in context: A337357 A340420 A127885 * A337150 A280234 A368383

Adjacent sequences: A006574 A006575 A006576 * A006578 A006579 A006580

Keyword

nonn,nice,easy,hear,look

Author

N. J. A. Sloane, Bill Gosper

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001

"Escape clause" added to definition by N. J. A. Sloane, Jun 06 2017

Status

approved