A198584 -id:A198584 - cp's OEIS frontend

A198585 The second odd number appearing in the Collatz sequence starting with A198584(n).

5, 5, 5, 85, 5, 341, 85, 5, 341, 85, 5, 341, 85, 5461, 5, 341, 21845, 85, 5461, 5, 341, 21845, 85, 5461, 5, 341, 21845, 85, 5461, 349525, 5, 341, 21845, 1398101, 85, 5461, 349525, 5, 341, 21845, 1398101, 85, 5461, 349525, 5, 341, 21845, 1398101, 85, 5461

Offset: 1

T. D. Noe, Oct 28 2011

nonn

A198586 has the unique numbers appearing in this sequence.

T. D. Noe, Table of n, a(n) for n = 1..1009

Cf. A079813, A198584, A198586.

a(n) = A079813(1 + (-1 + b(n))^2 + c(n))*(A198586(2*(1 + c(n))) - A198586(3 - 2*(b(n) - c(n)))) + A198586(3 - 2*(b(n) - c(n))) where b(n) = ceiling(sqrt(n)/sqrt(3)) and c(n) = (-1+n-3*(b(n)-1)^2) mod (2*b(n)-1). - Alan Michael Gómez Calderón, Jan 22 2025

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

A198586 a(n) = (4^A001651(n+1) - 1)/3: numbers (4^k-1)/3 for k > 1, not multiples of 3.

Original entry on oeis.org

5, 85, 341, 5461, 21845, 349525, 1398101, 22369621, 89478485, 1431655765, 5726623061, 91625968981, 366503875925, 5864062014805, 23456248059221, 375299968947541, 1501199875790165, 24019198012642645, 96076792050570581, 1537228672809129301

Offset: 1

T. D. Noe, Oct 30 2011

Numbers coprime to 6 producing 2 odd numbers in the Collatz iteration.
Numbers appearing in A198585 (sorted and duplicates removed). These numbers occur in A002450, numbers of the form (4^k-1)/3, for k = 2, 4, 5, 7, 8, 10, ... (note that k a multiple of 3 does not appear).
A124477 \ {0,1} is a subset: for these n, 3n+1 = 2^(p-3) with p > 3 prime, whence also n !== 0 (mod 3). - M. F. Hasler, Oct 16 2018
These are exactly the odd non-multiples of 3 such 3n+1 = 2^m for some m, i.e., n = (2^m-1)/3. This is possible iff m = 2k, so we get n = (4^k-1)/3. Then n == 0 (mod 3) <=> 4^k == 1 (mod 9) <=> k == 0 (mod 3) <=> k not in A001651. This yields the FORMULA. (Multiples of 3 are excluded because the original definition implied that the terms are in the Collatz-orbit of another odd number, i.e., of the form n = (3x+1)/2^r, which is impossible for x a multiple of 3.) - M. F. Hasler, Oct 16 2018
From Wolfdieter Lang, Jan 14 2022: (Start)
a(n) mod 8 = 5. As subsequence of A002450 for n >= 1.
{a(n) mod 6} == repeat{5, 1}. See the first comment, and the periodicity modulo 6 of A002450 for n >= 1.
{a(n) mod 72} == repeat{5, 13, 53, 61, 29, 37}. Proof by induction: First with the bisection formulas, a(1+2*k) = (4^(2+3*k) - 1)/3 and a(2+2*k) = (4^(3*k+4) - 1)/3, for k >= 0, then trisection, using (4^9 - 1)/3 = 873819 = 9*9709. (End)

T. D. Noe, Table of n, a(n) for n = 1..100
Index entries for linear recurrences with constant coefficients, signature (1,64,-64).

Cf. A001651, A002450, A124477, A198584.

[4^(3*n  div 2 + 1) div 3: n in [1..25]]; // Vincenzo Librandi, Oct 20 2018

e = 19; ex = Complement[Range[2,3*e], 3*Range[e]]; (4^ex - 1)/3
(* Second program: *)
Rest@ Map[(4^# - 1)/3 &, LinearRecurrence[{1, 1, -1}, {1, 2, 4}, 21]] (* Michael De Vlieger, Oct 17 2018 *)

is(n)=gcd(n,6)==1&&(n=3*n+1)>>valuation(n,2)==1 \\ M. F. Hasler, Oct 16 2018

A198586(n)=4^(3*n\2+1)\3 \\ M. F. Hasler, Oct 16 2018

Vec(x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)) + O(x^20)) \\ Colin Barker, Jan 17 2020

a(n) = (4^A001651(n+1) - 1)/3. - M. F. Hasler, Oct 16 2018
From Colin Barker, Jan 17 2020: (Start)
G.f.: x*(5 + 80*x - 64*x^2) / ((1 - x)*(1 - 8*x)*(1 + 8*x)).
a(n) = a(n-1) + 64*a(n-2) - 64*a(n-3) for n>3.
a(n) = (-1 + (-8)^n + 3*8^n) / 3.
(End)

Definition corrected by M. F. Hasler, Oct 16 2018

A228871 Odd numbers producing 3 out-of-order odd numbers in the Collatz (3x+1) iteration.

Original entry on oeis.org

3, 227, 14563, 932067, 59652323, 3817748707, 244335917283, 15637498706147, 1000799917193443, 64051194700380387, 4099276460824344803, 262353693492758067427, 16790636383536516315363, 1074600728546337044183267, 68774446626965570827729123

Offset: 1

T. D. Noe, Sep 12 2013

Sequence A198584 gives the first term of the Collatz sequence having exactly 3 odd numbers. This sequence is the subset of A198584 for which the second odd number is larger than the first. The second odd number is (2^(6*n - 2) - 1)/3, which always occurs as the third term of the sequence.

{a(n) mod 6} = {repeat(3, 5, 1)}, and a(n) mod 8 = 3 for all n. Proof from the formula of a(n) in terms of A198586 given below, using the modulo 72 congruence of the odd indexed part of A198586 given there. - Wolfdieter Lang, Jan 14 2022

			The number 3 has the Collatz iteration {3, 10, 5, 16, 8, 4, 2, 1}, which has three out-of-order odd numbers {3, 5, 1}.

Index entries for linear recurrences with constant coefficients, signature (65,-64).

Cf. A198584 (Collatz iterations having 3 odd numbers).
Cf. A228872 (Collatz iterations producing 3 in-order odd numbers).
Cf. A198586.

Table[(2*(2^(6*n - 2) - 1)/3 - 1)/3, {n, 15}]

a(n)=4^(3*n-1)\3*2\3 \\ Charles R Greathouse IV, Mar 11 2017

a(n) = (64^n/2 - 5)/9. - Alois P. Heinz, Dec 08 2021
From Wolfdieter Lang, Jan 12 2022: (Start)
a(n) = (2*A198586(2*n-1) - 1)/3. See the Mathematica program.
G.f.: x*(3 + 32*x)/((1 - x)*(1 - 64*x)). (End)

A198587 Odd numbers producing 4 odd numbers in the Collatz iteration.

Original entry on oeis.org

17, 35, 69, 75, 141, 151, 277, 301, 565, 605, 1109, 1137, 1205, 2261, 2275, 2417, 2421, 4437, 4549, 4821, 4835, 4849, 9045, 9101, 9669, 9685, 9699, 17749, 18197, 19285, 19341, 19397, 19417, 36181, 36405, 38677, 38741, 38797, 38833, 38835, 70997, 72789, 72817

Offset: 1

T. D. Noe, Oct 31 2011

nonn

Start with A385110. If k is in sequence then so is 4*k + 1. - Ralf Stephan, Jun 18 2025

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..287 (first 134 terms from Vincenzo Librandi)

Cf. A198584, A385110.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t = {}; Do[If[Length[Select[Collatz[n], OddQ]] == 4, AppendTo[t, n]], {n, 1, 100000, 2}]; t

A350053 a(n) = (2^(3*n + 3 + (-1)^n) - (6 + (-1)^n))/9, for n >= 1.

Original entry on oeis.org

3, 113, 227, 7281, 14563, 466033, 932067, 29826161, 59652323, 1908874353, 3817748707, 122167958641, 244335917283, 7818749353073, 15637498706147, 500399958596721, 1000799917193443, 32025597350190193, 64051194700380387

Offset: 1

Wolfdieter Lang, Jan 20 2022

Labels of nodes at level L = 1 of the Collatz tree with only odd numbers congruent to 1, 3, and 7 modulo 8, named here CToddr.
a(n) is given by the successor of the non-leaf node labels of the (reduced) Collatz tree with odd numbers (named here CTodd) at level 1 given by A198586(n), for n >= 1. See a comment in A347834 for the construction of CTodd. (For all labels of CTodd at level 1 see {A002450(k)}_{k>=2}.) The present sequence gives the labels of the (further) reduced rooted tree CToddr, at level L = 1. Level L = 0 has the root labeled 1, and this node has a directed 1-cycle.
The successor of a node label u of the tree CTodd is given by (4*u - 1)/3 if u == 1 (mod 6), (2*u - 1)/3 if u == 5 (mod 6), and there is no successor if the label u == 3 (mod 6) (a leaf).
This sequence is motivated by a draft of Immo O. Kerner (see A347834 and the link).
Sorted set of all A385109(A198584(i)), i>0 (conjectured but easy to see). - Ralf Stephan, Jun 18 2025

Winston de Greef, Table of n, a(n) for n = 1..1100
Index entries for linear recurrences with constant coefficients, signature (0,65,0,-64).

Cf. A002450, A198586, A228871, A347834, A350054.

a[n_] := (2^(3*n + 3 + (-1)^n) - (6 + (-1)^n))/9; Array[a, 20] (* Amiram Eldar, Jan 21 2022 *) (* or *)
LinearRecurrence[{0, 65, 0, -64}, {3, 113, 227, 7281}, 20] (* Georg Fischer, Sep 30 2022 *)

a(n) = (2^(3*n + 3 + (-1)^n))\9 \\ Winston de Greef, Jan 28 2024

Bisection: a(2*k-1) = (2^(6*k-1) - 5)/9 = A228871(k), a(2*k) = (4^(3*k+2) - 7)/9 = A350054(k), for k >= 1.
a(n) = (2^(3*n+ 2 + b(n)) - (5 + b(n)))/9, with b(n) = 1 + (-1)^n = A010673(n-1), for n >= 1. See the name.
G.f.: Bisection: x*(3 + 32*x)/((1 - x)*(1 - 64*x)) and x*(113 - 64*x)/((1 - x)*(1 - 64*x)).
G.f.: x*(3 + 113*x + 32*x^2 - 64*x^3)/((1 - x^2)*(1 - 64*x^2)).

A198593 Odd numbers producing 10 odd numbers in the Collatz iteration.

Original entry on oeis.org

43, 87, 89, 173, 177, 179, 349, 355, 357, 385, 423, 693, 705, 709, 717, 729, 761, 769, 771, 777, 847, 1393, 1397, 1411, 1415, 1421, 1429, 1457, 1459, 1465, 1481, 1523, 1539, 1541, 1547, 1555, 1569, 1689, 1693, 1697, 2773, 2787, 2801, 2821, 2831, 2837, 2869

Offset: 1

T. D. Noe, Oct 31 2011

nonn

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A198584.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t = {}; Do[If[Length[Select[Collatz[n], OddQ]] == 10, AppendTo[t, n]], {n, 1, 10000, 2}]; t

A228872 Odd numbers producing 3 decreasing odd numbers in the Collatz (3x+1) iteration.

Original entry on oeis.org

13, 53, 113, 213, 453, 853, 909, 1813, 3413, 3637, 7253, 7281, 13653, 14549, 29013, 29125, 54613, 58197, 58253, 116053, 116501, 218453, 232789, 233013, 464213, 466005, 466033, 873813, 931157, 932053, 1856853, 1864021, 1864133, 3495253, 3724629, 3728213

Offset: 1

T. D. Noe, Sep 12 2013

nonn

Sequence A198584 gives the first term of the Collatz sequence having exactly 3 odd numbers. This sequence is the subset of A198584 for which the second odd number is smaller than the first.

			The number 13 has the Collatz iteration {13, 40, 20, 10, 5, 16, 8, 4, 2, 1}, which has three odd numbers in decreasing order {13, 5, 1}.

Cf. A198584 (Collatz iterations having 3 odd numbers).

Cf. A228871 (Collatz iterations producing 3 out-of-order odd numbers).

donQ[n_]:=Module[{od=Differences[Select[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&], OddQ]]}, Length[ od] ==2&&Max[od]<0]; Select[Range[1,373*10^4,2],donQ] (* Harvey P. Dale, Sep 23 2019 *)

A351122 Irregular triangle read by rows in which row n lists the number of divisions by 2 after tripling steps in the Collatz 3x+1 trajectory of 2n+1 until it reaches 1.

Original entry on oeis.org

1, 4, 4, 1, 1, 2, 3, 4, 2, 1, 1, 2, 3, 4, 1, 2, 3, 4, 3, 4, 1, 1, 1, 5, 4, 2, 3, 4, 1, 3, 1, 2, 3, 4, 6, 1, 1, 5, 4, 2, 1, 3, 1, 2, 3, 4, 1, 2, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 2, 1, 1, 1, 2, 3, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 4, 2, 2, 4, 3, 1, 1, 5, 4, 3, 1, 2, 3, 4

Offset: 1

Flávio V. Fernandes, Feb 01 2022

			Triangle starts at T(1,0):
   n\k   0   1   2   3   4   5   6   7   8 ...
   1:    1   4
   2:    4
   3:    1   1   2   3   4
   4:    2   1   1   2   3   4
   5:    1   2   3   4
   6:    3   4
   7:    1   1   1   5   4
   8:    2   3   4
   9:    1   3   1   2   3   4
  10:    6
  11:    1   1   5   4
  12:    2   1   3   1   2   3   4
  13:    1   2   1   1   1   1   2   2   1   2   1   1   2  ... (see A372362)
  ...
For n=6, the trajectory of 2*n+1 = 13 is as follows. The tripling steps ("=>") are followed by runs of 3 and then 4 halvings ("->"), so row n=6 is 3, 4.
  13  =>  40 -> 20 -> 10 -> 5  =>  16 -> 8 -> 4 -> 2 -> 1
    triple   \------------/   triple  \---------------/
               3 halvings                4 halvings
Runs of halvings are divisions by 2^T(n,k). Row n=11 is 1, 1, 5, 4 and its steps starting from 2*n+1 = 23 reach 1 by a nested expression
  (((((((23*3+1)/2^1)*3+1)/2^1)*3+1)/2^5)*3+1)/2^4 = 1.

Cf. A075680 (row lengths), A166549 (row sums), A351123 (row partial sums).
Cf. A256598.
Cf. A020988 (where row is [2*n]).
Cf. A198584 (where row length is 2), A228871 (where row is [1, x]).
Cf. A372362 (row 13, the first 41 terms).

row(n) = my(m=2*n+1, list=List()); while (m != 1, if (m%2, m = 3*m+1, my(nb = valuation(m,2)); m/=2^nb; listput(list, nb));); Vec(list); \\ Michel Marcus, Jul 18 2022

T(n,k) = log_2( (3*A256598(n,k)+1) / A256598(n,k+1) ).

Corrected by Michel Marcus, Jul 18 2022

A198588 Odd numbers producing 5 odd numbers in the Collatz iteration.

Original entry on oeis.org

11, 23, 45, 93, 181, 201, 369, 373, 401, 403, 725, 739, 753, 803, 805, 1477, 1493, 1507, 1605, 1611, 1613, 2901, 2957, 3013, 3033, 3213, 3221, 3223, 5909, 5973, 6029, 6065, 6067, 6421, 6445, 6453, 6465, 11605, 11829, 12053, 12131, 12133, 12853, 12885, 12893

Offset: 1

T. D. Noe, Oct 31 2011

nonn

Vincenzo Librandi and T. D. Noe, Table of n, a(n) for n = 1..900 (terms 1 to 128 from Vincenzo Librandi)

Cf. A198584.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t = {}; Do[If[Length[Select[Collatz[n], OddQ]] == 5, AppendTo[t, n]], {n, 1, 20000, 2}]; t

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A198585 The second odd number appearing in the Collatz sequence starting with A198584(n).

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

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A198586 a(n) = (4^A001651(n+1) - 1)/3: numbers (4^k-1)/3 for k > 1, not multiples of 3.

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A228871 Odd numbers producing 3 out-of-order odd numbers in the Collatz (3x+1) iteration.

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A198587 Odd numbers producing 4 odd numbers in the Collatz iteration.

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A350053 a(n) = (2^(3*n + 3 + (-1)^n) - (6 + (-1)^n))/9, for n >= 1.

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A198593 Odd numbers producing 10 odd numbers in the Collatz iteration.

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A228872 Odd numbers producing 3 decreasing odd numbers in the Collatz (3x+1) iteration.

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