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

Author

Joseph L. Pe, Dec 20 2002

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

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

Cf. A258145, A003602.

Programs

Haskell

a078719 =
   (+ 1) . length . filter odd . takeWhile (> 2) . (iterate a006370)
a078719_list = map a078719 [1..]
-- Reinhard Zumkeller, Oct 08 2011

Maple

a:= proc(n) option remember; `if`(n<2, 1,
      `if`(n::even, a(n/2), 1+a(3*n+1)))
    end:
seq(a(n), n=1..94);  # Alois P. Heinz, Jan 17 2025

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 *)

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

Formula

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

a(n) = A258145(A003602(n)-1). - Alan Michael Gómez Calderón, Sep 15 2024

Extensions

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