A062053 - OEIS

A062053

Numbers with 3 odd integers in their Collatz (or 3x+1) trajectory.

3, 6, 12, 13, 24, 26, 48, 52, 53, 96, 104, 106, 113, 192, 208, 212, 213, 226, 227, 384, 416, 424, 426, 452, 453, 454, 768, 832, 848, 852, 853, 904, 906, 908, 909, 1536, 1664, 1696, 1704, 1706, 1808, 1812, 1813, 1816, 1818, 3072, 3328, 3392, 3408, 3412, 3413, 3616

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OFFSET

1,1

COMMENTS

The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd (A006370).

The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.

A078719(a(n)) = 3; A006667(a(n)) = 2.

REFERENCES

J. R. Goodwin, Results on the Collatz Conjecture, Sci. Ann. Comput. Sci. 13 (2003) pp. 1-16

J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

LINKS

Reinhard Zumkeller and David A. Corneth, Table of n, a(n) for n = 1..16191 (first 250 terms from Reinhard Zumkeller, terms < 10^25)

J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

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 2-automatic sequences.

FORMULA

The two formulas giving this sequence are listed in Corollary 3.1 and Corollary 3.2 in J. R. Goodwin with the following caveats: the value x cannot equal zero in Corollary 3.2, one must multiply the formulas by all powers of 2 (2^1, 2^2, ...) to get the evens. - Jeffrey R. Goodwin, Oct 26 2011

EXAMPLE

The Collatz trajectory of 3 is (3,10,5,16,8,4,2,1), which contains 3 odd integers.

MATHEMATICA

Collatz[n_?OddQ] := (3n + 1)/2; Collatz[n_?EvenQ] := n/2; oddIntCollatzCount[n_] := Length[Select[NestWhileList[Collatz, n, # != 1 &], OddQ]]; Select[Range[4000], oddIntCollatzCount[#] == 3 &] (* Alonso del Arte, Oct 28 2011 *)

PROG

(Haskell)

import Data.List (elemIndices)

a062053 n = a062053_list !! (n-1)

a062053_list = map (+ 1) $ elemIndices 3 a078719_list

-- Reinhard Zumkeller, Oct 08 2011

CROSSREFS

Cf. A006370, A078719, A006667.

Cf. A000079, A062052, A062054, A062055, A062056, A062057, A062058, A062059, A062060.

Cf. A198584 (this sequence without the even numbers).