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A006370
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Image of n under the `3x+1' map.
(Formerly M3198)
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65
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4, 1, 10, 2, 16, 3, 22, 4, 28, 5, 34, 6, 40, 7, 46, 8, 52, 9, 58, 10, 64, 11, 70, 12, 76, 13, 82, 14, 88, 15, 94, 16, 100, 17, 106, 18, 112, 19, 118, 20, 124, 21, 130, 22, 136, 23, 142, 24, 148, 25, 154, 26, 160, 27, 166, 28, 172, 29, 178, 30, 184, 31, 190, 32, 196, 33
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,1
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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 an unsolved problem. It is conjectured that the answer is yes.
The Krasikov-Lagarias paper shows that at least N^.84 of the positive numbers <N fall into the 4-2-1 cycle of the 3x+1 problem. This is far short of what we think is true, that all positive numbers fall into this cycle, but it is a step. - Richard Schroeppel, May 01, 2002
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REFERENCES
| R. K. Guy, Unsolved Problems in Number Theory, E16.
J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arithmetica, LVI (1990), pp. 33-53.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010.
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..1000
I. Krasikov and J. C. Lagarias, Bounds for the 3x+1 Problem using Difference Inequalities
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
E. Roosendaal, On the 3x+1 problem
S. Plouffe, Approximations de S\'{e}ries G\'{e}n\'{e}ratrices et Quelques Conjectures, Dissertation, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
S. Plouffe, 1031 Generating Functions and Conjectures, Universit\'{e} du Qu\'{e}bec \`{a} Montr\'{e}al, 1992.
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences related to linear recurrences with constant coefficients, signature (0,2,0,-1).
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FORMULA
| G.f.: (4x+x^2 +2x^3) / (1-x^2)^2.
a(n) = (1/4)(7n+2-(-1)^n(5n+2)) - Benoit Cloitre, May 12 2002
a(n) = ((n mod 2)*2 + 1)*n/(2 - (n mod 2)) + (n mod 2). - Reinhard Zumkeller, Sep 12 2002
a(n) = A014682(n+1)*A000034(n). [From R. J. Mathar, Mar 09 2009]
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MAPLE
| f := n-> if n mod 2 = 0 then n/2 else 3*n+1; fi;
A006370:=(4+z+2*z**2)/(z-1)**2/(1+z)**2; [S. Plouffe in his 1992 dissertation.]
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PROG
| (PARI) for(n=1, 100, print1((1/4)*(7*n+2-(-1)^n*(5*n+2)), ", "))
(PARI) A006370(n)=if(n%2, 3*n+1, n/2) [From Michael B. Porter, May 29 2010]
(Haskell)
a006370 n | m /= 0 = 3 * n + 1
| otherwise = n' where (n', m) = divMod n 2
-- Reinhard Zumkeller, Oct 07 2011
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CROSSREFS
| Cf. A139391, A016945, A005408, A016825, A082286.
Cf. A070165.
Sequence in context: A059926 A138775 A121529 * A108759 A158824 A039806
Adjacent sequences: A006367 A006368 A006369 * A006371 A006372 A006373
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KEYWORD
| nonn,nice,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001
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