

A062054


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


4



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
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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.
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


REFERENCES

J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182185.


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. 116.
J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182185.
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 2automatic sequences.


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


EXAMPLE

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


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 *)


PROG

(Haskell)
import Data.List (elemIndices)
a062054 n = a062054_list !! (n1)
a062054_list = map (+ 1) $ elemIndices 4 a078719_list
 Reinhard Zumkeller, Oct 08 2011


CROSSREFS

Cf. A000079, A006370, A062052, A062053, A062055, A062056, A062057, A062058, A062059, A062060, A092893, A198587.
Sequence in context: A168579 A135637 A040272 * A164008 A013577 A234942
Adjacent sequences: A062051 A062052 A062053 * A062055 A062056 A062057


KEYWORD

nonn,easy


AUTHOR

David W. Wilson


STATUS

approved



