A062060 -id:A062060 - cp's OEIS frontend

A006667 Number of tripling steps to reach 1 from n in '3x+1' problem, or -1 if 1 is never reached.

0, 0, 2, 0, 1, 2, 5, 0, 6, 1, 4, 2, 2, 5, 5, 0, 3, 6, 6, 1, 1, 4, 4, 2, 7, 2, 41, 5, 5, 5, 39, 0, 8, 3, 3, 6, 6, 6, 11, 1, 40, 1, 9, 4, 4, 4, 38, 2, 7, 7, 7, 2, 2, 41, 41, 5, 10, 5, 10, 5, 5, 39, 39, 0, 8, 8, 8, 3, 3, 3, 37, 6, 42, 6, 3, 6, 6, 11, 11, 1, 6, 40, 40, 1, 1, 9, 9, 4, 9, 4, 33, 4, 4, 38

Offset: 1

N. J. A. Sloane and Bill Gosper

A075680, which gives the values for odd n, isolates the essential behavior of this sequence. - T. D. Noe, Jun 01 2006

A033959 and A033958 give record values and where they occur. - Reinhard Zumkeller, Jan 08 2014

J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 204, Problem 22.
R. K. Guy, Unsolved Problems in Number Theory, E16.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..10000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Eric Weisstein's World of Mathematics, Collatz Problem.
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Equals A078719(n)-1.

Cf. A000079, A000265, A006370, A006577, A006666 (halving steps), A092893, A127789, A139391, A209229.

a006667 = length . filter odd . takeWhile (> 2) . (iterate a006370)
a006667_list = map a006667 [1..]
-- Reinhard Zumkeller, Oct 08 2011

a:= proc(n) option remember; `if`(n<2, 0,
      `if`(n::even, a(n/2), 1+a(3*n+1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 08 2023

Table[Count[Differences[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]], ?Positive], {n,100}] (* _Harvey P. Dale, Nov 14 2011 *)

for(n=2,100,s=n; t=0; while(s!=1,if(s%2==0,s=s/2,s=(3*s+1)/2; t++); if(s==1,print1(t,","); ); ))

def a(n):
    if n==1: return 0
    x=0
    while True:
        if n%2==0: n/=2
        else:
            n = 3*n + 1
            x+=1
        if n<2: break
    return x
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 14 2017

a(1) = 0, a(n) = a(n/2) if n is even, a(n) = a(3n+1)+1 if n>1 is odd. The Collatz conjecture is that this defines a(n) for all n >= 1.
a(n) = A078719(n) - 1; a(A000079(n))=0; a(A062052(n))=1; a(A062053(n))=2; a(A062054(n))=3; a(A062055(n))=4; a(A062056(n))=5; a(A062057(n))=6; a(A062058(n))=7; a(A062059(n))=8; a(A062060(n))=9. - Reinhard Zumkeller, Oct 08 2011
a(n*2^k) = a(n), for all k >= 0. - L. Edson Jeffery, Aug 11 2014
a(n) = floor(log(2^A006666(n)/n)/log(3)). - Joe Slater, Aug 30 2017
a(n) = a(A085062(n)) + A007814(n+1) for n >= 2. - Alan Michael Gómez Calderón, Feb 07 2025
From  Alan Michael Gómez Calderón, Mar 31 2025: (Start)
a(n) = a(A139391(n)) + (n mod 2) for n >= 2;
a(n) = a(A139391(A000265(n))) - A209229(n) + 1 for n >= 2;
a(n) = a(A000265(A139391(n))) + (n mod 2) for n >= 2. (End)

More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001

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

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.

Original entry on oeis.org

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

Joseph L. Pe, Dec 20 2002

nonn

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

			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.

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

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

Cf. A258145, A003602.

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

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

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 *)

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

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

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

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

A062052 Numbers with exactly 2 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

5, 10, 20, 21, 40, 42, 80, 84, 85, 160, 168, 170, 320, 336, 340, 341, 640, 672, 680, 682, 1280, 1344, 1360, 1364, 1365, 2560, 2688, 2720, 2728, 2730, 5120, 5376, 5440, 5456, 5460, 5461, 10240, 10752, 10880, 10912, 10920, 10922, 20480, 21504, 21760, 21824

Offset: 1

David W. Wilson

nonn

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.
The sequence consists of terms of A002450 and their 2^k multiples. The first odd integer in the trajectory is one of the terms of A002450 and the second odd one is the terminal 1. - Antti Karttunen, Feb 21 2006
This sequence looks to appear first in the literature on page 1285 in R. E. Crandall.

			The Collatz trajectory of 5 is (5,16,8,4,2,1), which contains 2 odd integers.

Reinhard Zumkeller and T. D. Noe, Table of n, a(n) for n = 1..1000 (first 100 terms from Reinhard Zumkeller)
R. E. Crandall, On the 3x+1 problem, Math. Comp., 32 (1978) 1281-1292.
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.

Cf. A062053, A062054, A062055, A062056, A062057, A062058, A062059, A062060, A122196, A122197.
Is this a subset of A115774?
Column k=2 of A354236.

import Data.List (elemIndices)
a062052 n = a062052_list !! (n-1)
a062052_list = map (+ 1) $ elemIndices 2 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[22000], countOdd[Collatz[#]] == 2 &] (* T. D. Noe, Dec 03 2012 *)

for(n=2,100000,s=n; t=0; while(s!=1,if(s%2==0,s=s/2,s=3*s+1; t++); if(s*t==1,print1(n,","); ); ))

def a(n):
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        if n not in l:
            l.append(n)
            if n<2: break
        else: break
    return len([i for i in l if i % 2])
print([n for n in range(1, 22001) if a(n)==2]) # Indranil Ghosh, Apr 14 2017

A078719(a(n)) = 2; A006667(a(n)) = 1.

a(n) = 2^x * (4^y - 1)/3 where x = A122196(n) - 1 and y = A122197(n) + 1. - Alan Michael Gómez Calderón, Jan 16 2025 after Antti Karttunen

A354236 A(n,k) is the n-th number m such that the Collatz (or 3x+1) trajectory starting at m contains exactly k odd integers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

Original entry on oeis.org

1, 5, 2, 3, 10, 4, 17, 6, 20, 8, 11, 34, 12, 21, 16, 7, 22, 35, 13, 40, 32, 9, 14, 23, 68, 24, 42, 64, 25, 18, 15, 44, 69, 26, 80, 128, 33, 49, 19, 28, 45, 70, 48, 84, 256, 43, 65, 50, 36, 29, 46, 75, 52, 85, 512, 57, 86, 66, 51, 37, 30, 88, 136, 53, 160, 1024

Offset: 1

Alois P. Heinz, May 20 2022

			Square array A(n,k) begins:
    1,   5,  3,  17, 11,  7,  9,  25,  33,  43, ...
    2,  10,  6,  34, 22, 14, 18,  49,  65,  86, ...
    4,  20, 12,  35, 23, 15, 19,  50,  66,  87, ...
    8,  21, 13,  68, 44, 28, 36,  51,  67,  89, ...
   16,  40, 24,  69, 45, 29, 37,  98, 130, 172, ...
   32,  42, 26,  70, 46, 30, 38,  99, 131, 173, ...
   64,  80, 48,  75, 88, 56, 72, 100, 132, 174, ...
  128,  84, 52, 136, 90, 58, 74, 101, 133, 177, ...
  256,  85, 53, 138, 92, 60, 76, 102, 134, 178, ...
  512, 160, 96, 140, 93, 61, 77, 196, 260, 179, ...

Alois P. Heinz, Rows n = 1..150, flattened
Wikipedia, Collatz Conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for sequences that are permutations of the positive integers

Columns k=1-12 give: A011782, A062052, A062053, A062054, A062055, A062056, A062057, A062058, A062059, A062060, A072466, A072122.
Row n=1 gives A092893(k-1).
Main diagonal gives A380244.
Cf. A006577, A006667, A078719.

b:= proc(n) option remember; irem(n, 2, 'r')+
      `if`(n=1, 0, b(`if`(n::odd, 3*n+1, r)))
    end:
A:= proc() local h, p, q; p, q:= proc() [] end, 0;
      proc(n, k)
        if k=1 then return 2^(n-1) fi;
        while nops(p(k))

b[n_] := b[n] = Module[{q, r}, {q, r} = QuotientRemainder[n, 2]; r +
     If[n == 1, 0, b[If[OddQ[n], 3*n + 1, q]]]];
A = Module[{h, p, q}, p[_] = {}; q = 0;
     Function[{n, k}, If[k == 1, 2^(n - 1)];
     While[Length[p[k]] < n, q = q + 1;
        h = b[q];
        p[h] = Append[p[h], q]];
     p[k][[n]]]];
Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 12}] // Flatten (* Jean-François Alcover, Jun 02 2022, after Alois P. Heinz *)

A078719(A(n,k)) = k.

A062053 Numbers with exactly 3 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

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

David W. Wilson

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.

			The Collatz trajectory of 3 is (3,10,5,16,8,4,2,1), which contains 3 odd integers.

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.

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.

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.

import Data.List (elemIndices)
a062053 n = a062053_list !! (n-1)
a062053_list = map (+ 1) $ elemIndices 3 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

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 *)

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

A062055 Numbers with 5 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

11, 22, 23, 44, 45, 46, 88, 90, 92, 93, 176, 180, 181, 184, 186, 201, 352, 360, 362, 368, 369, 372, 373, 401, 402, 403, 704, 720, 724, 725, 736, 738, 739, 744, 746, 753, 802, 803, 804, 805, 806, 1408, 1440, 1448, 1450, 1472, 1476, 1477, 1478, 1488, 1492, 1493, 1506

Offset: 1

David W. Wilson

nonn

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)) = 5; A006667(a(n)) = 4.

			The Collatz trajectory of 11 is (11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 5 odd integers.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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.

Cf. A062052-A062060.

Column k=5 of A354236.

import Data.List (elemIndices)
a062055 n = a062055_list !! (n-1)
a062055_list = map (+ 1) $ elemIndices 5 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[2000], countOdd[Collatz[#]] == 5 &] (* T. D. Noe, Dec 03 2012 *)

A062056 Numbers with 6 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

7, 14, 15, 28, 29, 30, 56, 58, 60, 61, 112, 116, 117, 120, 122, 224, 232, 234, 240, 241, 244, 245, 267, 448, 464, 468, 469, 480, 482, 483, 488, 490, 497, 534, 535, 537, 896, 928, 936, 938, 960, 964, 965, 966, 976, 980, 981, 985, 994, 995, 1068, 1069, 1070, 1073

Offset: 1

David W. Wilson

nonn

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)) = 6; A006667(a(n)) = 5.

			The Collatz trajectory of 7 is (7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 6 odd integers.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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.

Cf. A062052-A062060.

Column k=6 of A354236.

import Data.List (elemIndices)
a062056 n = a062056_list !! (n-1)
a062056_list = map (+ 1) $ elemIndices 6 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[1000], countOdd[Collatz[#]] == 6 &] (* T. D. Noe, Dec 03 2012 *)

A062057 Numbers with 7 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

9, 18, 19, 36, 37, 38, 72, 74, 76, 77, 81, 144, 148, 149, 152, 154, 162, 163, 288, 296, 298, 304, 308, 309, 321, 324, 325, 326, 331, 576, 592, 596, 597, 608, 616, 618, 625, 642, 643, 648, 650, 652, 653, 662, 663, 713, 715, 1152, 1184, 1192, 1194, 1216, 1232, 1236, 1237

Offset: 1

David W. Wilson

nonn

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)) = 7; A006667(a(n)) = 6.

			The Collatz trajectory of 9 is (9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 7 odd integers.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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.

Cf. A062052-A062060.

Column k=7 of A354236.

import Data.List (elemIndices)
a062057 n = a062057_list !! (n-1)
a062057_list = map (+ 1) $ elemIndices 7 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[1000], countOdd[Collatz[#]] == 7 &] (* T. D. Noe, Dec 03 2012 *)

A062058 Numbers with 8 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

25, 49, 50, 51, 98, 99, 100, 101, 102, 196, 197, 198, 200, 202, 204, 205, 217, 392, 394, 396, 397, 400, 404, 405, 408, 410, 433, 434, 435, 441, 475, 784, 788, 789, 792, 794, 800, 808, 810, 816, 820, 821, 833, 857, 866, 867, 868, 869, 870, 875, 882, 883, 950, 951, 953

Offset: 1

David W. Wilson

nonn

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)) = 8; A006667(a(n)) = 7.

			The Collatz trajectory of 25 is (25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 8 odd integers.

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

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
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.

Cf. A062052-A062060.

Column k=8 of A354236.

import Data.List (elemIndices)
a062058 n = a062058_list !! (n-1)
a062058_list = map (+ 1) $ elemIndices 8 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[1000], countOdd[Collatz[#]] == 8 &] (* T. D. Noe, Dec 03 2012 *)

A062059 Numbers with 9 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

33, 65, 66, 67, 130, 131, 132, 133, 134, 260, 261, 262, 264, 266, 268, 269, 273, 289, 520, 522, 524, 525, 528, 529, 532, 533, 536, 538, 546, 547, 555, 571, 577, 578, 579, 583, 633, 635, 1040, 1044, 1045, 1048, 1050, 1056, 1058, 1059, 1064, 1066, 1072, 1076, 1077

Offset: 1

David W. Wilson

nonn

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)) = 9; A006667(a(n)) = 8.

			The Collatz trajectory of 33 is (33, 100, 50, 25, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 9 odd integers.

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

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

Cf. A062052, A062053, A062054, A062055, A062056, A062057, A062058, A062060.

Column k=9 of A354236.

import Data.List (elemIndices)
a062059 n = a062059_list !! (n-1)
a062059_list = map (+ 1) $ elemIndices 9 a078719_list
-- Reinhard Zumkeller, Oct 08 2011

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countOdd[lst_] := Length[Select[lst, OddQ]]; Select[Range[1000], countOdd[Collatz[#]] == 9 &] (* T. D. Noe, Dec 03 2012 *)

def a(n):
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        if n not in l:
            l+=[n, ]
            if n<2: break
        else: break
    return len([i for i in l if i%2])
[n for n in range(30, 1101) if a(n)==9] # Indranil Ghosh, Apr 14 2017

cp's OEIS Frontend

A006667 Number of tripling steps to reach 1 from n in '3x+1' problem, or -1 if 1 is never reached.

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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.

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A062052 Numbers with exactly 2 odd integers in their Collatz (or 3x+1) trajectory.

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A354236 A(n,k) is the n-th number m such that the Collatz (or 3x+1) trajectory starting at m contains exactly k odd integers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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A062053 Numbers with exactly 3 odd integers in their Collatz (or 3x+1) trajectory.

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A062055 Numbers with 5 odd integers in their Collatz (or 3x+1) trajectory.

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