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A006577
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Number of halving and tripling steps to reach 1 in `3x+1' problem.
(Formerly M4323)
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63
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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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.
a(n) = A112695(n) + 2 for n > 2. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 18 2008
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REFERENCES
| R. K. Guy, Unsolved Problems in Number Theory, E16.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.
J. L. Simons, On the nonexistence of 2-cycles for the 3x+1 problem, Math. Comp. 75 (2005), 1565-1572.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
| N. J. A. Sloane, Table of n, a(n) for n = 1..10000
Geometry.net, Links on Collatz Problem
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, PlanetMath.org, Collatz problem
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]
Mathematical BBS, Biblography on Collatz Sequence
P. Picart, Algorithme de Collatz et conjecture de Syracuse
E. Roosendaal, On the 3x+1 problem
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
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FORMULA
| a(n) = A006666(n) + A006667(n)
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EXAMPLE
| a(5)=5 because the trajectory of 5 is (5,16,8,4,2,1).
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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!}] (*From Vladimir Joseph Stephan Orlovsky (4vladimir(AT)gmail.com), Jan 08 2011*)
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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)
Contribution from Michael B. Porter (michael_b_porter(AT)yahoo.com), Jun 05 2010: (Start)
(PARI) step(n)=if(n%2, 3*n+1, n/2)
A006577(n)=if(n==1, 0, A006577(step(n))+1) (End)
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CROSSREFS
| See A070165 for triangle giving trajectories of n = 1, 2, 3, ....
Cf. A125731, A127885, A127886, A008908, A112695.
A008884, A161021 ,A161022, A161023. [From Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Jun 03 2009]
Sequence in context: A200237 A072761 A127885 * A073652 A117029 A128475
Adjacent sequences: A006574 A006575 A006576 * A006578 A006579 A006580
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), R. W. Gosper
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001
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