A062054 Numbers with 4 odd integers in their Collatz (or 3x+1) trajectory.

17, 34, 35, 68, 69, 70, 75, 136, 138, 140, 141, 150, 151, 272, 276, 277, 280, 282, 300, 301, 302, 544, 552, 554, 560, 564, 565, 600, 602, 604, 605, 1088, 1104, 1108, 1109, 1120, 1128, 1130, 1137, 1200, 1204, 1205, 1208, 1210, 2176, 2208, 2216, 2218, 2240

Offset: 1

Author

David W. Wilson

Keywords

• nonn
• easy

Comments

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

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

A078719(a(n)) = 4; A006667(a(n)) = 3.

Numbers m such that (s0 - 4s1)/2m = 1 where s0 is the sum of the even elements and s1 the sum of the odd elements in the Collatz trajectory of m. - Michel Lagneau, Aug 13 2018

If m is in the sequence then so is 2*m, so one would only have to check odd numbers. - David A. Corneth, Aug 13 2018

Examples

			The Collatz trajectory of 17 is (17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 4 odd integers. - _Jeffrey R. Goodwin_, Oct 26 2011
		

References

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

Links

• David A. Corneth, Table of n, a(n) for n = 1..15549 (first 750 terms from Reinhard Zumkeller, terms < 10^15)
• 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.
• 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.

Crossrefs

Cf. A000079, A006370, A062052, A062053, A062055, A062056, A062057, A062058, A062059, A062060, A092893, A198587.

Column k=4 of A354236.

Programs

Haskell

import Data.List (elemIndices)
a062054 n = a062054_list !! (n-1)
a062054_list = map (+ 1) $ elemIndices 4 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Mathematica

col4Q[n_]:=Module[{c=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&]}, Count[c,?OddQ]==4]; Select[Range[2500],col4Q]  (* _Harvey P. Dale, Mar 21 2011 *)

Formula

The twelve formulas giving this sequence are listed in Corollary 3.3 in J. R. Goodwin with the following caveats: the value x cannot equal zero in formulas (3.16) and (3.20), one must multiply the formulas by all powers of 2 (2^1, 2^2, ...) to get the evens. - Jeffrey R. Goodwin, Oct 26 2011