A078719 Number of odd terms among n, f(n), f(f(n)), ...., 1 for the Collatz function (that is, until reaching "1" for the first time), or -1 if 1 is never reached.

1, 1, 3, 1, 2, 3, 6, 1, 7, 2, 5, 3, 3, 6, 6, 1, 4, 7, 7, 2, 2, 5, 5, 3, 8, 3, 42, 6, 6, 6, 40, 1, 9, 4, 4, 7, 7, 7, 12, 2, 41, 2, 10, 5, 5, 5, 39, 3, 8, 8, 8, 3, 3, 42, 42, 6, 11, 6, 11, 6, 6, 40, 40, 1, 9, 9, 9, 4, 4, 4, 38, 7, 43, 7, 4, 7, 7, 12, 12, 2, 7, 41, 41, 2, 2, 10, 10, 5, 10, 5, 34, 5, 5, 39

Offset

1,3

Comments

The Collatz function (related to the "3x+1 problem") is defined by: f(n) = n/2 if n is even; f(n) = 3n + 1 if n is odd. A famous conjecture states that n, f(n), f(f(n)), .... eventually reaches 1.

a(n) = A006667(n) + 1; a(A000079(n))=1; a(A062052(n))=2; a(A062053(n))=3; a(A062054(n))=4; a(A062055(n))=5; a(A062056(n))=6; a(A062057(n))=7; a(A062058(n))=8; a(A062059(n))=9; a(A062060(n))=10. - Reinhard Zumkeller, Oct 08 2011

The count includes also the starting value n if it is odd. See A286380 for the version which never includes n itself. - Antti Karttunen, Aug 10 2017

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Chris K. Caldwell and G. L. Honaker, Jr., Prime curio for 41 (which says 41 is a fixed point)

Eric Weisstein's World of Mathematics, Collatz Problem

Wikipedia, Collatz conjecture

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

Formula

a(n) = A286380(n) + A000035(n). - Antti Karttunen, Aug 10 2017

Example

The terms n, f(n), f(f(n)), ...., 1 for n = 12 are: 12, 6, 3, 10, 5, 16, 8, 4, 2, 1, of which 3 are odd. Hence a(12) = 3.

Mathematica

f[n_] := Module[{a, i, o}, i = n; o = 1; a = {}; While[i > 1, If[Mod[i, 2] == 1, o = o + 1]; a = Append[a, i]; i = f[i]]; o]; Table[f[i], {i, 1, 100}]
Table[Count[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &], _?OddQ], {n, 94}] (* Jayanta Basu, Jun 15 2013 *)

Prog

(Haskell)
a078719 =
   (+ 1) . length . filter odd . takeWhile (> 2) . (iterate a006370)
a078719_list = map a078719 [1..]
-- Reinhard Zumkeller, Oct 08 2011
(PARI) a(n) = {my(x=n, v=List([])); while(x>1, if(x%2==0, x=x/2, listput(v, x); x=3*x+1)); 1+#v; } \\ Jinyuan Wang, Dec 29 2019

Crossrefs

Cf. A006370, A014682, A078720, A139391, A286380.

Sequence in context: A138881 A070983 A078350 * A087227 A060477 A345037

Adjacent sequences: A078716 A078717 A078718 * A078720 A078721 A078722

Keyword

nonn

Author

Joseph L. Pe, Dec 20 2002

Extensions

"Escape clause" added to definition by N. J. A. Sloane, Jun 06 2017

Status

approved