

A006577


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


236



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
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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
Jason Holt, Loglog 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), 323.
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. 253266.
J. C. Lagarias, The 3x+1 Problem: an annotated bibliography, II (20002009), arXiv:0608208 [math.NT], 20062012.
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 2cycles for the 3x+1 problem, Math. Comp. 75 (2005), 15651572.
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


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 A073652
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



