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Search: id:A006370
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| A006370 |
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Image of n under the `3x+1' map.
(Formerly M3198)
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+0
45
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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; listen)
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OFFSET
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1,1
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COMMENT
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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
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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.
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.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
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.
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.
Index entries for sequences related to 3x+1 (or Collatz) problem
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FORMULA
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G.f.: (4x+x^2 +2x^3 ) / (1-x^2 )^2.
a(n)=(1/4)(7n+2-(-1)^n(5n+2)) - Benoit Cloitre (benoit7848c(AT)orange.fr), May 12 2002
a(n) = ((n mod 2)*2 + 1)*n/(2 - (n mod 2)) + (n mod 2). - Reinhard Zumkeller (reinhard.zumkeller(AT)lhsystems.com), Sep 12 2002
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MAPLE
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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; [Conjectured by S. Plouffe in his 1992 dissertation.]
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PROGRAM
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(PARI) for(n=1, 100, print1((1/4)*(7*n+2-(-1)^n*(5*n+2)), ", "))
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CROSSREFS
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Cf. A139391, A016945, A005408, A016825, A082286.
Sequence in context: A059926 A138775 A121529 this_sequence A108759 A039806 A030320
Adjacent sequences: A006367 A006368 A006369 this_sequence A006371 A006372 A006373
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KEYWORD
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nonn,nice
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AUTHOR
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njas
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EXTENSIONS
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More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001
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