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A006370 Image of n under the '3x+1' map.
(Formerly M3198)
100
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; text; internal format)
OFFSET

1,1

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^0.84 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

Also A016957 and A000027 interleaved. - Omar E. Pol, Jan 16 2014

a(n) is the image of a(2*n) under the 3*x+1 map. - L. Edson Jeffery, Aug 17 2014

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.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

T. D. Noe, Table of n, a(n) for n=1..1000

Darrell Cox, The 3n + 1 Problem: A Probabilistic Approach, Journal of Integer Sequences, Vol. 15 (2012), #12.5.2.

I. Krasikov and J. C. Lagarias, Bounds for the 3x+1 Problem using Difference Inequalities, arXiv:math/0205002 [math.NT], 2002.

J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.

J. C. Lagarias, The set of rational cycles for the 3x+1 problem, Acta Arithmetica, LVI (1990), pp. 33-53.

J. C. Lagarias, The 3x+1 Problem: An Annotated Bibliography (1963-2000), arXiv:math/0309224 [math.NT], 2003-2011.

J. C. Lagarias, The 3x+1 Problem: an annotated bibliography, II (2000-2009), arXiv:math/0608208 [math.NT], 2006-2012.

E. Roosendaal, On the 3x+1 problem

Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992.

Simon Plouffe, 1031 Generating Functions and Conjectures, Université du Québec à Montréal, 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 linear recurrences with constant coefficients, signature (0,2,0,-1).

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). - R. J. Mathar, Mar 09 2009

a(n) = a(a(2*n)) = -A001281(-n) for all n in Z. - Michael Somos, Nov 10 2016

E.g.f.: (2 + x)*sinh(x)/2 + 3*x*cosh(x). - Ilya Gutkovskiy, Dec 20 2016

EXAMPLE

G.f. = 4*x + x^2 + 10*x^3 + 2*x^4 + 16*x^5 + 3*x^6 + 22*x^7 + 4*x^8 + 28*x^9 + ...

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; # Simon Plouffe in his 1992 dissertation; uses offset 0

MATHEMATICA

f[n_]:=If[EvenQ[n], n/2, 3n+1]; Table[f[n], {n, 50}] (* Geoffrey Critzer, Jun 29 2013 *)

LinearRecurrence[{0, 2, 0, -1}, {4, 1, 10, 2}, 70] (* Harvey P. Dale, Jul 19 2016 *)

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) \\ 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

(Python)

def A006370(n):

....q, r = divmod(n, 2)

....return 3*n+1 if r else q # Chai Wah Wu, Jan 04 2015

(MAGMA) [(1/4)*(7*n+2-(-1)^n*(5*n+2)): n in [1..70]]; // Vincenzo Librandi, Dec 20 2016

CROSSREFS

Cf. A139391, A016945, A005408, A016825, A082286, A070165.

A006577 gives number of steps to reach 1.

Cf. A001281.

Sequence in context: A138775 A209385 A121529 * A262370 A108759 A158824

Adjacent sequences:  A006367 A006368 A006369 * A006371 A006372 A006373

KEYWORD

nonn,nice,easy

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001

STATUS

approved

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Last modified June 24 11:40 EDT 2017. Contains 288697 sequences.