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

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).

See also A198587.

Column k=3 of A354236.

Sequence in context: A116625 A287560 A318934 * A274652 A339552 A102040

Adjacent sequences: A062050 A062051 A062052 * A062054 A062055 A062056

Keyword

nonn,easy

Author

David W. Wilson

Status

approved