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A006667
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Number of tripling steps to reach 1 in `3x+1' problem.
(Formerly M0019)
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23
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0, 0, 2, 0, 1, 2, 5, 0, 6, 1, 4, 2, 2, 5, 5, 0, 3, 6, 6, 1, 1, 4, 4, 2, 7, 2, 41, 5, 5, 5, 39, 0, 8, 3, 3, 6, 6, 6, 11, 1, 40, 1, 9, 4, 4, 4, 38, 2, 7, 7, 7, 2, 2, 41, 41, 5, 10, 5, 10, 5, 5, 39, 39, 0, 8, 8, 8, 3, 3, 3, 37, 6, 42, 6, 3, 6, 6, 11, 11, 1, 6, 40, 40, 1, 1, 9, 9, 4, 9, 4, 33, 4, 4, 38
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| A075680, which gives the values for odd n, isolates the essential behavior of this sequence. - T. D. Noe (noe(AT)sspectra.com), Jun 01 2006
a(n) = A078719(n) - 1; a(A000079(n))=0; a(A062052(n))=1; a(A062053(n))=2; a(A062054(n))=3; a(A062055(n))=4; a(A062056(n))=5; a(A062057(n))=6; a(A062058(n))=7; a(A062059(n))=8; a(A062060(n))=9. [Reinhard Zumkeller, Oct 08 2011]
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REFERENCES
| J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 204, Problem 22.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..10000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
| a(1) = 0, a(n) = a(n/2) if n is even, a(n) = a(3n+1)+1 if n>1 is odd. The Collatz conjecture is that this defines a(n) for all n >= 1.
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MATHEMATICA
| Table[Count[Differences[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]], _?Positive], {n, 100}] (* From Harvey P. Dale, Nov 14 2011 *)
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PROG
| (PARI) for(n=2, 100, s=n; t=0; while(s!=1, if(s%2==0, s=s/2, s=(3*s+1)/2; t++); if(s==1, print1(t, ", "); ); ))
(Haskell)
a006667 = length . filter odd . takeWhile (> 2) . (iterate a006370)
a006667_list = map a006667 [1..]
-- Reinhard Zumkeller, Oct 08 2011
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CROSSREFS
| Equals A078719(n)-1.
Cf. A006370.
Sequence in context: A025247 A127767 A055509 * A112570 A127755 A180662
Adjacent sequences: A006664 A006665 A006666 * A006668 A006669 A006670
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KEYWORD
| nonn,nice
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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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