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A006667
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Number of tripling steps to reach 1 in '3x+1' problem, or -1 if 1 is never reached.
(Formerly M0019)
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48
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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
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OFFSET
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1,3
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COMMENTS
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A075680, which gives the values for odd n, isolates the essential behavior of this sequence. - T. D. Noe, 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
A033959 and A033958 give record values and where they occur. - Reinhard Zumkeller, Jan 08 2014
a(n*2^k) = a(n), for all k >= 0. - L. Edson Jeffery, Aug 11 2014
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REFERENCES
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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
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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, Collatz Problem.
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
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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.
a(n) = floor(log(2^A006666(n)/n)/log(3)). - Joe Slater, Aug 30 2017
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MATHEMATICA
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Table[Count[Differences[NestWhileList[If[EvenQ[#], #/2, 3#+1]&, n, #>1&]], _?Positive], {n, 100}] (* Harvey P. Dale, Nov 14 2011 *)
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PROG
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(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
(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 xrange(1, 101)] # Indranil Ghosh, Apr 14 2017
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CROSSREFS
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Equals A078719(n)-1.
Cf. A006370.
Cf. A006666 (halfing steps), A127789 (record indices of 2^A006666(n)/(3^a(n)*n)).
Sequence in context: A127767 A292577 A055509 * A112570 A127755 A180662
Adjacent sequences: A006664 A006665 A006666 * A006668 A006669 A006670
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KEYWORD
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nonn,nice
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AUTHOR
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N. J. A. Sloane, Bill Gosper
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EXTENSIONS
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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
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STATUS
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approved
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