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A075677 Reduced Collatz function R applied to the odd integers: a(n) = R(2n-1), where R(k) = (3k+1)/2^r, with r as large as possible. 9
1, 5, 1, 11, 7, 17, 5, 23, 13, 29, 1, 35, 19, 41, 11, 47, 25, 53, 7, 59, 31, 65, 17, 71, 37, 77, 5, 83, 43, 89, 23, 95, 49, 101, 13, 107, 55, 113, 29, 119, 61, 125, 1, 131, 67, 137, 35, 143, 73, 149, 19, 155, 79, 161, 41, 167, 85, 173, 11, 179, 91, 185, 47, 191, 97, 197 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The even-indexed terms a(2i+2) = 6i+5 = A016969(i), i >= 0 [Comment corrected by Bob Selcoe, Apr 06 2015]. The odd-indexed terms terms are the same as A067745. Note that this sequence is A016789 with all factors of 2 removed from each term. Also note that a(4i-1) = a(i). No multiple of 3 is in this sequence. See A075680 for the number of iterations of R required to yield 1.

From Bob Selcoe, Apr 06 2015 (Start):

All numbers in this sequence appear infinitely often.

From Eq. 1 and Eq. 2 in Formulas: Eq. 1 is used with 1/3 of the numbers in this sequence, Eq. 2 is used with 2/3 of the numbers.

(End)

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; see p. 57, also (90-9), p. 306.

LINKS

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

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

Index entries for sequences related to 3x+1 (or Collatz) problem

FORMULA

a(n) = A000265(6*n-2). - Reinhard Zumkeller, Jan 08 2014

From Bob Selcoe, Apr 05 2015 (Start):

For j>=0, the following two equations make a single function for all k:

Eq. 1: a(n) = (3n-1)/2^(2j+1) when k = ((4^(j+1)-1)/3) mod 2^(2j+3). Alternatively: a(n) = A016789(n-1)/A081294(j+1) when k = A002450(j+1) mod A081294(j+2).

Eq. 2: a(n) = (3n-1)/4^j when k = (5*2^(2j+1) - 1)/3) mod 4^(j+1).  Alternatively: a(n) = A016789(n-1)/A000302(j) when k = A072197(j) mod A000302(j+1).

(End)

a(n) = a(n + g*2^r) - 6*g. - Bob Selcoe, Apr 06 2015

EXAMPLE

a(11) = 1 because 21 is the 11th odd number and R(21) = 64/64 = 1.

From  Bob Selcoe, Apr 05 2015 (Start):

From Eq. 1: n=51; k=101 == 5 mod 32, j=1. a(51) = 152/8 = 19.

From Eq. 2: n=91; k=181 == 53 mod 64, j=2.  a(91) = 272/16 = 17.

(End)

n=59; a(59)=11, r=5. g = -1: 11 = a(27)=5 + 6; g = 1: 11 = a(91)=17 - 6; g=2: 11 = a(123)=23 - 2*6; g=3: 11 = a(155)=29 - 3*6; etc. - Bob Selcoe, Apr 06 2015

MAPLE

f:=proc(n) local t1;

if n=1 then RETURN(1) else

t1:=3*n+1;

while t1 mod 2 = 0 do t1:=t1/2; od;

RETURN(t1); fi;

end;

(from N. J. A. Sloane, Jan 21 2011)

MATHEMATICA

nextOddK[n_] := Module[{m=3n+1}, While[EvenQ[m], m=m/2]; m]; (* assumes odd n *) Table[nextOddK[n], {n, 1, 200, 2}]

PROG

(PARI) a(n)=n+=2*n-1; n>>valuation(n, 2) \\ Charles R Greathouse IV, Jul 05 2013

(Haskell)

a075677 = a000265 . subtract 2 . (* 6) -- Reinhard Zumkeller, Jan 08 2014

CROSSREFS

Cf. A000302, A002450, A016789, A016969, A065677, A072197, A075680, A081294.

Sequence in context: A131782 A242060 A185953 * A051853 A159074 A147414

Adjacent sequences:  A075674 A075675 A075676 * A075678 A075679 A075680

KEYWORD

easy,nonn

AUTHOR

T. D. Noe, Sep 25 2002

STATUS

approved

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Last modified April 28 10:16 EDT 2015. Contains 257094 sequences.