%I A006577 M4323
%S A006577 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,
%T A006577 18,18,18,106,5,26,13,13,21,21,21,34,8,109,8,29,16,16,16,104,11,24,24,
%U A006577 24,11,11,112,112,19,32,19,32,19,19,107,107,6,27,27,27,14,14,14,102,22
%N A006577 Number of halving and tripling steps to reach 1 in `3x+1' problem.
%C A006577 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.
%C A006577 a(n) = A112695(n) + 2 for n > 2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Apr 18 2008
%D A006577 R. K. Guy, Unsolved Problems in Number Theory, E16.
%D A006577 J. L. Simons, On the nonexistence of 2-cycles for the 3x+1 problem, Math. 
               Comp. 75 (2005), 1565-1572.
%D A006577 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A006577 N. J. A. Sloane, <a href="b006577.txt">Table of n, a(n) for n = 1..10000</
               a>
%H A006577 Geometry.net, <a href="http://www.geometry.net/theorems_and_conjectures/
               collatz_problem.html">Links on Collatz Problem</a>
%H A006577 Jason Holt, <a href="a006577_1Blog.png">Log-log plot of first billion 
               terms</a>
%H A006577 Jason Holt, <a href="a006577_1B.png">Plot of 1 billion values of the 
               number of steps to drop below n</a> (A060445), log scale on x axis
%H A006577 Jason Holt, <a href="a006577_10B.png">Plot of 10 billion values of the 
               number of steps to drop below n</a> (A060445), log scale on x axis
%H A006577 A. Krowne, PlanetMath.org, <a href="http://planetmath.org/encyclopedia/
               CollatzProblem.html">Collatz problem</a>
%H A006577 J. C. Lagarias, <a href="http://www.cecm.sfu.ca/organics/papers/lagarias/
               paper/html/paper.html">The 3x+1 problem and its generalizations</
               a>, Amer. Math. Monthly, 92 (1985), 3-23.
%H A006577 J. C. Lagarias, <a href="http://www.g4g4.com/contentsmmpp.html">How random 
               are 3x+1 function iterates?</a>, 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.
%H A006577 Mathematical BBS, <a href="http://felix.unife.it/Root/d-Mathematics/d-Number-theory/
               b-3x+1">Biblography on Collatz Sequence</a>
%H A006577 P. Picart, <a href="http://trucsmaths.free.fr/js_syracuse.htm">Algorithme 
               de Collatz et conjecture de Syracuse</a>
%H A006577 E. Roosendaal, <a href="http://www.ericr.nl/wondrous/index.html">On the 
               3x+1 problem</a>
%H A006577 G. Villemin's Almanach of Numbers, <a href="http://translate.google.com/
               translate?hl=en&sl=fr&u=http://membres.lycos.fr/villemingerard/Iteration/
               Syracuse.htm#top">Cycle of Syracuse</a>
%H A006577 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               CollatzProblem.html">Collatz Problem</a>
%H A006577 Wikipedia, <a href="http://en.wikipedia.org/wiki/Collatz_conjecture">
               Collatz Conjecture</a>
%H A006577 <a href="Sindx_3.html#3x1">Index entries for sequences related to 3x+1 
               (or Collatz) problem</a>
%F A006577 a(n) = A006666(n) + A006667(n)
%e A006577 a(5)=5 because the trajectory of 5 is (5,16,8,4,2,1).
%o A006577 (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)
%Y A006577 See A070165 for triangle giving trajectories of n = 1, 2, 3, ....
%Y A006577 Cf. A125731, A127885, A127886, A008908, A112695.
%Y A006577 A008884, A161021 ,A161022, A161023. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), 
               Jun 03 2009]
%Y A006577 Sequence in context: A074457 A072761 A127885 this_sequence A073652 A117029 
               A128475
%Y A006577 Adjacent sequences: A006574 A006575 A006576 this_sequence A006578 A006579 
               A006580
%K A006577 nonn,nice,easy
%O A006577 1,3
%A A006577 N. J. A. Sloane (njas(AT)research.att.com), R. W. Gosper
%E A006577 More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001