A220237 -id:A220237 - cp's OEIS frontend

A070165 Irregular triangle read by rows giving trajectory of n in Collatz problem.

1, 2, 1, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 5, 16, 8, 4, 2, 1, 6, 3, 10, 5, 16, 8, 4, 2, 1, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 8, 4, 2, 1, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 10, 5, 16, 8, 4, 2, 1, 11, 34, 17, 52, 26, 13

Offset: 1

Eric W. Weisstein, Apr 23 2002

n-th row has A008908(n) entries (unless some n never reaches 1, in which case the triangle ends with an infinite row). [Escape clause added by N. J. A. Sloane, Jun 06 2017]
A216059(n) is the smallest number not occurring in n-th row; see also A216022.
Comment on the mp3 file from Gordon Charlton (the recording artist Beat Frequency). The piece uses the first 3242 terms (i.e. the first 100 hailstone sequences), with pitch modulus 36, duration modulus 2. Its musicality stems from the many repetitions and symmetries within the sequence, and in particular the infrequency of multiples of 3. This means that when the pitch modulus is a multiple of 12 the notes are predominantly in the symmetric octatonic scale, known to modern classical composers as the second of Messiaen's modes of limited transposition, and to jazz musicians as half-whole diminished. - N. J. A. Sloane, Jan 30 2019

			The irregular array a(n,k) starts:
n\k   0  1  2  3  4   5  6   7  8  9 10 11 12 13 14 15 16 17 18 19
1:    1
2:    2  1
3:    3 10  5 16  8   4  2   1
4:    4  2  1
5:    5 16  8  4  2   1
6:    6  3 10  5 16   8  4   2  1
7:    7 22 11 34 17  52 26  13 40 20 10  5 16  8  4  2  1
8:    8  4  2  1
9:    9 28 14  7 22  11 34  17 52 26 13 40 20 10  5 16  8  4  2  1
10:  10  5 16  8  4   2  1
11:  11 34 17 52 26  13 40  20 10  5 16  8  4  2  1
12:  12  6  3 10  5  16  8   4  2  1
13:  13 40 20 10  5  16  8   4  2  1
14:  14  7 22 11 34  17 52  26 13 40 20 10  5 16  8  4  2  1
15:  15 46 23 70 35 106 53 160 80 40 20 10  5 16  8  4  2  1
... Reformatted and extended by _Wolfdieter Lang_, Mar 20 2014

T. D. Noe, Rows n = 1..100 of triangle, flattened
Gordon Charlton ("Beat Frequency"), Hailstone Trajectory (mp3 file)
David Eisenbud and Brady Haran, UNCRACKABLE? The Collatz Conjecture, Numberphile Video, 2016.
David Rabahy, Hailstone Sequence presented as a spreadsheet
Anatoly E. Voevudko, File of first 10K Collatz sequences
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370 (step), A008908 (row lengths), A033493 (row sums).
Cf. A220237 (sorted rows), A347270 (array), A192719.
Cf. A070168 (Terras triangle), A256598 (reduced triangle).
Cf. A254311, A257480 (and crossrefs therein).
Cf. A280408 (primes).

a070165 n k = a070165_tabf !! (n-1) !! (k-1)
a070165_tabf = map a070165_row [1..]
a070165_row n = (takeWhile (/= 1) $ iterate a006370 n) ++ [1]
a070165_list = concat a070165_tabf
-- Reinhard Zumkeller, Oct 07 2011

T:= proc(n) option remember; `if`(n=1, 1,
      [n, T(`if`(n::even, n/2, 3*n+1))][])
    end:
seq(T(n), n=1..15);  # Alois P. Heinz, Jan 29 2021

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Flatten[Table[Collatz[n], {n, 10}]] (* T. D. Noe, Dec 03 2012 *)

row(n, lim=0)={if (n==1, return([1])); my(c=n, e=0, L=List(n)); if(lim==0, e=1; lim=n*10^6); for(i=1, lim, if(c%2==0, c=c/2, c=3*c+1); listput(L, c); if(e&&c==1, break)); return(Vec(L)); } \\ Anatoly E. Voevudko, Mar 26 2016; edited by Michel Marcus, Aug 10 2021

def a(n):
    if n==1: return [1]
    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 l
for n in range(1, 101): print(a(n)) # Indranil Ghosh, Apr 14 2017

T(n,k) = T^{(k)}(n) with the k-th iterate of the Collatz map T with T(n) = 3*n+1 if n is odd and T(n) = n/2 if n is even, n >= 1. T^{(0)}(n) = n. k = 0, 1, ..., A008908(n) - 1. - Wolfdieter Lang, Mar 20 2014

Name specified and row length A-number corrected by Wolfdieter Lang, Mar 20 2014

A025586 Largest value in '3x+1' trajectory of n.

Original entry on oeis.org

1, 2, 16, 4, 16, 16, 52, 8, 52, 16, 52, 16, 40, 52, 160, 16, 52, 52, 88, 20, 64, 52, 160, 24, 88, 40, 9232, 52, 88, 160, 9232, 32, 100, 52, 160, 52, 112, 88, 304, 40, 9232, 64, 196, 52, 136, 160, 9232, 48, 148, 88, 232, 52, 160, 9232, 9232, 56, 196, 88, 304, 160, 184, 9232

Offset: 1

David W. Wilson

Here by definition the trajectory ends when 1 is reached. Therefore this sequence differs for n = 1 and n = 2 from A056959, which considers the orbit ending in the infinite loop 1 -> 4 -> 2 -> 1.
a(n) = A220237(n,A006577(n)). - Reinhard Zumkeller, Jan 03 2013
A006885 and A006884 give record values and where they occur. - Reinhard Zumkeller, May 11 2013
For n > 2, a(n) is divisible by 4. See the explanatory comment in A056959. - Peter Munn, Oct 14 2019
In an email of Aug 06 2023, Guy Chouraqui observes that the digital root of a(n) appears to be either 7 or a multiple of 4 for all n > 2.  (See also A006885.) - N. J. A. Sloane, Aug 11 2023

			The 3x + 1 trajectory of 9 is 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 (see A033479). Since the largest number in that sequence is 52, a(9) = 52.

T. D. Noe, Table of n, a(n) for n = 1..10000
Christian Hercher, There are no Collatz m-Cycles with m <= 91, J. Int. Seq. (2023) Vol. 26, Article 23.3.5.
Philippe Picart, Algorithme de Collatz et conjecture de Syracuse
Index entries for sequences related to 3x+1 (or Collatz) problem

Essentially the same as A056959: only a(1) and a(2) differ, see Comments.

Cf. A006370, A006577, A006884, A006885, A220237.

a025586 = last . a220237_row
-- Reinhard Zumkeller, Jan 03 2013, Aug 29 2012

a:= proc(n) option remember; `if`(n=1, 1,
      max(n, a(`if`(n::even, n/2, 3*n+1))))
    end:
seq(a(n), n=1..87);  # Alois P. Heinz, Oct 16 2021

collatz[a0_Integer, maxits_:1000] := NestWhileList[If[EvenQ[#], #/2, 3# + 1] &, a0, Unequal[#, 1, -1, -10, -34] &, 1, maxits]; (* collatz[n] function definition by Eric Weisstein *) Flatten[Table[Take[Sort[Collatz[n], Greater], 1], {n, 60}]] (* Alonso del Arte, Nov 14 2007 *)
collatzMax[n_] := Module[{r = m = n}, While[m > 2, If[OddQ[m], m = 3 * m + 1; If[m > r, r = m], m = m/2]]; r]; Table[ collatzMax[n], {n, 100}] (* Jean-François Alcover, Jan 28 2015, after Charles R Greathouse IV *)
(* Using Weisstein's collatz[n] definition above *) Table[Max[collatz[n]], {n, 100}] (* Alonso del Arte, May 25 2019 *)

a(n)=my(r=n);while(n>2,if(n%2,n=3*n+1;if(n>r,r=n),n/=2));r \\ Charles R Greathouse IV, Jul 19 2011

def a(n):
    if n<2: return 1
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        if not n in l:
            l+=[n, ]
            if n<2: break
        else: break
    return max(l)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 14 2017

def collatz(n: Int): Int = (n % 2) match {
  case 0 => n / 2
  case 1 => 3 * n + 1
}
def collatzTrajectory(start: Int): List[Int] = if (start == 1) List(1)
else {
  import scala.collection.mutable.ListBuffer
  var curr = start; var trajectory = new ListBuffer[Int]()
  while (curr > 1) { trajectory += curr; curr = collatz(curr) }
  trajectory.toList
}
for (n <- 1 to 100) yield collatzTrajectory(n).max // Alonso del Arte, Jun 02 2019

A192719 Chain of Collatz sequences.

Original entry on oeis.org

1, 2, 1, 3, 10, 5, 16, 8, 4, 2, 1, 6, 3, 10, 5, 16, 8, 4, 2, 1, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 12, 6, 3, 10, 5, 16, 8, 4, 2, 1

Offset: 1

Robert C. Lyons, Dec 31 2012

The sequence is a chain of Collatz sequences. The first Collatz sequence in the chain is (1). Each of the subsequent Collatz sequences in the chain starts with the minimum positive integer that does not appear in the previous Collatz sequences. If the Collatz conjecture is true, then each Collatz sequence in the chain will end with 1, and the chain will include an infinite number of distinct Collatz sequences. If the Collatz conjecture is false, then the chain will end with the first Collatz sequence that does not converge to 1.

T(n,1) = A177729(n). - Reinhard Zumkeller, Jan 03 2013

			The first Collatz sequence in the chain is (1). The second Collatz sequence in the chain is (2, 1), which starts with 2, since 2 is the smallest positive integer that doesn't appear the first Collatz sequence. The third Collatz sequence in the chain is (3, 10, 5, 16, 8, 4, 2, 1), which starts with 3, since 3 is the smallest positive integer that doesn't appear the previous Collatz sequences.
Thus this irregular array starts:
1;
2,  1;
3, 10,  5, 16,  8,  4,  2,  1;
6,  3, 10,  5, 16,  8,  4,  2,  1;
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10,  5, 16,  8, 4,  2, 1;
9, 28, 14,  7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Robert C. Lyons, Chain of Collatz sequences generator program in Java
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A070167, A070165, A061641, A177729, A192719.

Cf. A220263 (row lengths); A070165, A220237.

a192719 n k = a192719_tabf !! (n-1) !! (k-1)
a192719_row n = a192719_tabf !! (n-1)
a192719_tabf = f [1..] where
   f (x:xs) = (a070165_row x) : f (del xs $ a220237_row x)
   del us [] = us
   del us'@(u:us) vs'@(v:vs) | u > v     = del us' vs
                             | u < v     = u : del us vs'
                             | otherwise = del us vs
-- Reinhard Zumkeller, Jan 03 2013

Java
```
See Lyons link.
```

A005184 Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k.

Original entry on oeis.org

31, 83, 293, 347, 671, 19151, 2025797

Offset: 1

N. J. A. Sloane

The Collatz sequence of a number k is defined as a(1)=k, a(j+1) = a(j)/2 if a(j) is even, 3*a(j) + 1 if a(j) is odd.
No others less than 250000000. - Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 07 2006
There are no more terms < 10^11. - Donovan Johnson, Nov 28 2013
There are no more terms < 10^15. - Alun Stokes, Mar 01 2021

			The Collatz sequence of 31 is 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310 (see A008884) ... 310 is a multiple of 31, so the number 31 is "self-contained".

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).

Alexander Rahn, Max Henkel, Sourangshu Ghosh, Eldar Sultanow, and Idriss Aberkane, An algorithm for linearizing Collatz convergence, hal-03286608 [math.DS], 2021.
Eldar Sultanow, Christian Koch, and Sean Cox, Collatz Sequences in the Light of Graph Theory, Universität Potsdam (Germany, 2020).
Index entries for sequences related to 3x+1 (or Collatz) problem

The ratios "higher multiple of k" / k are given in A059198.

Cf. A070165, A220237, A303698.

T:= proc(n) option remember; `if`(n=1, 1,
      [n, T(`if`(n::even, n/2, 3*n+1))][])
    end:
q:= n-> ormap(x-> x>n and irem(x, n)=0, [T(n)]):
select(q, [2*i-1$i=1..10000])[];  # Alois P. Heinz, Aug 04 2025

isSelfContained[n_] := Module[{d}, d = n; While[d != 1, If[EvenQ[d], d = d/2, d = 3 * d + 1]; If[IntegerQ[d/n], Return[True]]]; Return[False]]; For[n = 1, n <= 250000000, n += 2, If[isSelfContained[n], Print[n]]]; (* Sam Handler (sam_5_5_5_0(AT)yahoo.com), Aug 07 2006 *)
scnQ[n_] := MemberQ[Divisible[#, n] & / @Rest[NestWhileList[If[EvenQ[#], #/2, 3# + 1] &, n, # > 1 &]], True]; Select[Range[1, 2100001, 2], scnQ] (* Harvey P. Dale, Oct 21 2011 *)

m=5; d=2; while(1,n=(3*m+1)\2; until(n==1,n=if(n%2,3*n+1,n\2); if(n%m==0,print(m," ",n); break)); m+=d; d=6-d)

More terms from Robert G. Wilson v

Better description from Jack Brennen, Feb 07 2003

A246436 Number of numbers 1,...,n not occurring in the Collatz trajectory starting with n.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 4, 2, 4, 4, 3, 6, 4, 8, 11, 8, 4, 8, 12, 15, 10, 14, 13, 11, 16, 17, 12, 15, 19, 21, 26, 16, 22, 26, 16, 19, 21, 21, 31, 28, 34, 23, 28, 31, 34, 34, 36, 29, 29, 33, 40, 43, 36, 40, 36, 33, 39, 37, 44, 47, 45, 48, 57, 42, 41, 45, 53, 56

Offset: 1

Reinhard Zumkeller, Sep 01 2014

nonn

a(n) = n - A159999(n).

			8-4-2-1: a(8) = #{3,5,6,7} = 4;
9-28-14-7-22-11-34-17-52-26-13-40-20-10-5-16-8-4-2-1: a(9) = #{3,6} = 2;
10-5-16-8-4-2-1: a(10) = #{3,6,7,9} = 4;
11-34-17-52-26-13-40-20-10-5-16-8-4-2-1: a(11) = #{3,6,7,9} = 4;
12-6-3-10-5-16-8-4-2-1: a(12) = #{7,9,11} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A159999, A220237, A070165.

import Data.List ((\\), genericIndex)
a246436 n = length $ [1..n] \\ genericIndex a220237_tabf (n - 1)

Table[Length[Complement[Range[n],NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]],{n,70}] (* Harvey P. Dale, May 27 2018 *)

A225105 Odd numbers n such that the largest odd term in Collatz(3x+1) trajectory of n is prime.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 35, 37, 39, 53, 59, 61, 67, 75, 79, 87, 89, 99, 101, 105, 113, 115, 119, 131, 149, 153, 157, 173, 179, 181, 187, 197, 211, 219, 229, 241, 247, 249, 255, 267, 269, 277, 281, 307, 317, 329, 349, 371, 373, 383, 397

Offset: 1

Jayanta Basu, Apr 28 2013

nonn

			15 is in the list since highest odd number in Collatz trajectory of 15 is 53, a prime.

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

Cf. A087232.

Cf. A010051, A070165, A005408, A220237, A025586.

a225105 n = a225105_list !! (n-1)
a225105_list = filter
   ((== 1) . a010051' . maximum . filter odd . a070165_row) a005408_list
-- Reinhard Zumkeller, Apr 30 2013

Coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3*# + 1] &,n, #>1&];t={};Do[If[PrimeQ[Max[Select[Coll[n],OddQ]]],AppendTo[t,n]],{n,1,300,2}];t

A376079 a(n) is the largest difference of adjacent elements in the sorted list of the Collatz trajectory elements of n.

Original entry on oeis.org

1, 6, 2, 8, 6, 12, 4, 12, 6, 12, 4, 20, 12, 54, 8, 14, 12, 30, 6, 32, 12, 54, 8, 18, 14, 1944, 12, 36, 54, 1944, 16, 18, 12, 54, 12, 56, 30, 66, 20, 1944, 22, 48, 8, 68, 54, 1944, 24, 38, 18, 78, 14, 80, 1944, 1944, 12, 24, 30, 66, 54, 54, 1944, 1944, 32, 48, 12

Offset: 2

Markus Sigg, Sep 09 2024

Trajectories end when they reach 1.

			The trajectory of 3 is (3,10,5,16,8,4,2,1), the sorted list of the trajectory elements is (1,2,3,4,5,8,10,16), the list of differences is (1,1,1,1,3,2,6) with maximum 6, so a(3) = 6.

Markus Sigg, Table of n, a(n) for n = 2..10001

Cf. A220237 (sorted trajectory), A008908, A025586.

Table[Max[Differences[Sort[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]]],{n,2,70}] (* Harvey P. Dale, Aug 24 2025 *)

a(n) = my(L = List([n])); while(n > 1, n = if(n % 2 == 0, n/2, 3*n + 1); listput(L, n)); listsort(L); vecmax(vector(#L - 1, i, L[i+1] - L[i]));

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A070165 Irregular triangle read by rows giving trajectory of n in Collatz problem.

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A025586 Largest value in '3x+1' trajectory of n.

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A192719 Chain of Collatz sequences.

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A005184 Self-contained numbers: odd numbers k whose Collatz sequence contains a higher multiple of k.

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A246436 Number of numbers 1,...,n not occurring in the Collatz trajectory starting with n.

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A225105 Odd numbers n such that the largest odd term in Collatz(3x+1) trajectory of n is prime.

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A376079 a(n) is the largest difference of adjacent elements in the sorted list of the Collatz trajectory elements of n.

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