A006667 -id:A006667 - cp's OEIS frontend

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.

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.

A062060 Numbers with 10 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

43, 86, 87, 89, 172, 173, 174, 177, 178, 179, 344, 346, 348, 349, 354, 355, 356, 357, 358, 385, 423, 688, 692, 693, 696, 698, 705, 708, 709, 710, 712, 714, 716, 717, 729, 761, 769, 770, 771, 777, 846, 847, 1376, 1384, 1386, 1392, 1393, 1396, 1397, 1410, 1411, 1415

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.
The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.
A078719(a(n)) = 10; A006667(a(n)) = 9.

			The Collatz trajectory of 43 is (43, 130, 65, 196, 98, 49, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which contains 10 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-A062059.

Column k=10 of A354236.

import Data.List (elemIndices)
a062060 n = a062060_list !! (n-1)
a062060_list = map (+ 1) $ elemIndices 10 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[#]] == 10 &] (* 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.append(n)
            if n<2: break
        else: break
    return len([1 for i in l if i%2])
print([n for n in range(40, 1501) if a(n)==10]) # Indranil Ghosh, Apr 14 2017

A070167 a(n) is the smallest starting value that produces a Collatz sequence in which n occurs.

Original entry on oeis.org

1, 2, 3, 3, 3, 6, 7, 3, 9, 3, 7, 12, 7, 9, 15, 3, 7, 18, 19, 7, 21, 7, 15, 24, 25, 7, 27, 9, 19, 30, 27, 21, 33, 7, 15, 36, 37, 25, 39, 7, 27, 42, 43, 19, 45, 15, 27, 48, 43, 33, 51, 7, 15, 54, 55, 37, 57, 19, 39, 60, 27, 27, 63, 21, 43, 66, 39, 45, 69, 15, 27, 72, 73, 43, 75, 25

Offset: 1

Eric W. Weisstein, Apr 23 2002

nonn

a(n) <= n. - Robert G. Wilson v, Jan 14 2015

T. D. Noe, Table of n, a(n) for n=1..10000
Eric Weisstein's World of Mathematics, Collatz Sequence
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006667, A070165.

Cf. A010120, A054646.

import Data.List (findIndex)
import Data.Maybe (fromJust)
a070167 n = fromJust (findIndex (elem n) a070165_tabf) + 1
-- Reinhard Zumkeller, Jan 02 2013

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 100; t = Table[0, {nn}]; n = 0; zeros = nn; While[zeros > 0, n++; c = Collatz[n]; Do[If[i <= nn && t[[i]] == 0, t[[i]] = n; zeros--], {i, c}]]; t (* T. D. Noe, Dec 03 2012 *)

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

A092893 Smallest starting value in a Collatz '3x+1' sequence such that the sequence contains exactly n tripling steps.

Original entry on oeis.org

1, 5, 3, 17, 11, 7, 9, 25, 33, 43, 57, 39, 105, 135, 185, 123, 169, 219, 159, 379, 283, 377, 251, 167, 111, 297, 395, 263, 175, 233, 155, 103, 137, 91, 121, 161, 107, 71, 47, 31, 41, 27, 73, 97, 129, 171, 231, 313, 411, 543, 731, 487, 327, 859, 1145, 763, 1017, 1351

Offset: 0

Hugo Pfoertner, Mar 11 2004

nonn

First occurrence of n in A006667.
These are the odd (primitive) terms in A129304. - T. D. Noe, Apr 09 2007
Except for a(1) = 5, all values are congruent {1, 3, 7} (mod 8). Reason: If n is 5 (mod 8) then the Collatz trajectory starting with m = (n - 1)/4 contains the same number of tripling steps, because n = 4m + 1 and the Collatz 3x + 1 step results in 3*(4m + 1) + 1 = 12m + 4 which gets reduced by halving to 3m + 1, without changing the number of tripling steps. - Ralf Stephan, Jun 19 2025

			a(4)=11 because the Collatz sequence 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 is the first sequence containing 4 tripling steps.

T. D. Noe, Table of n, a(n) for n=0..300
Jeffrey R. Goodwin, The 3x+1 Problem and Integer Representations, Arxiv preprint arXiv:1504.03040 [math.NT], 2015.
Eric Weisstein's World of Mathematics, Collatz Problem
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006667, A033491, A092892, A087228.

Row n=1 of A354236.

a[n_]:=Length[Select[NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &],OddQ]]; Table[i=1; While[a[i]!=n,i=i+2]; i,{n,58}] (* Jayanta Basu, May 27 2013 *)

cp's OEIS Frontend

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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A062060 Numbers with 10 odd integers in their Collatz (or 3x+1) trajectory.

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A070167 a(n) is the smallest starting value that produces a Collatz sequence in which n occurs.

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

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

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