A006884 -id:A006884 - cp's OEIS frontend

A006885 Record highest point of trajectory before reaching 1 in '3x+1' problem, corresponding to starting values in A006884.

1, 2, 16, 52, 160, 9232, 13120, 39364, 41524, 250504, 1276936, 6810136, 8153620, 27114424, 50143264, 106358020, 121012864, 593279152, 1570824736, 2482111348, 2798323360, 17202377752, 24648077896, 52483285312, 56991483520, 90239155648, 139646736808

Offset: 1

Robert Munafo

Both the 3x+1 steps and the halving steps are counted.
Record values in A025586: a(n) = A025586(A006884(n)) and A025586(m) < a(n) for m < A006884(n). - Reinhard Zumkeller, May 11 2013
In an email of Aug 06 2023, Guy Chouraqui observes that the digital root of a(n) appears to be 7 for all n > 2.  - N. J. A. Sloane, Aug 11 2023

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 96.
D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.
G. T. Leavens and M. Vermeulen, 3x+1 search problems, Computers and Mathematics with Applications, 24 (1992), 79-99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..97, using data from Eric Rosendaal's 3x+1 Path records (terms 1..84 from T. D. Noe)
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Peter Munn, Plot2 graph of record highest point against starting value
Eric Roosendaal, 3x+1 Path Records
Robert G. Wilson v, Letter to N. J. A. Sloane with attachments, Jan. 1989
Robert G. Wilson v, Tables of A6877, A6884, A6885, Jan. 1989
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A006877, A006878, A033492.

a006885 = a025586 . a006884  -- Reinhard Zumkeller, May 11 2013

mcoll[n_]:=Max@@NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>=n&]; t={1,max=2}; Do[If[(y=mcoll[n])>max,AppendTo[t,max=y]],{n,3,10^6,4}]; t (* Jayanta Basu, May 28 2013 *)

A060410 In the '3x+1' problem, take the sequence of starting values which set new records for the highest point of the trajectory before reaching 1 (A006884); sequence gives associated maximal value reached in the trajectory with that start.

Original entry on oeis.org

2, 8, 26, 80, 4616, 6560, 19682, 20762, 125252, 638468, 3405068, 4076810, 13557212, 25071632, 53179010, 60506432, 296639576, 785412368, 1241055674, 1399161680, 8601188876, 12324038948, 26241642656, 28495741760

Offset: 1

N. J. A. Sloane, Apr 06 2001

nonn

David Barina, Table of n, a(n) for n = 1..93 (first 88 terms from the web page of Tomás Oliveira e Silva)
David Barina, Path records.
Tomás Oliveira e Silva, Tables (gives many more terms)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A060409, A060411.

A060409 In the '3x+1' problem, take the sequence of starting values which set new records for the highest point of the trajectory before reaching 1 (A006884); sequence gives associated "dropping time", number of steps to reach a lower value than the start.

Original entry on oeis.org

1, 4, 7, 7, 59, 13, 40, 23, 81, 61, 70, 65, 54, 72, 65, 59, 127, 105, 110, 59, 72, 164, 140, 73, 170, 105, 149, 97, 135, 183, 99, 99, 124, 156, 200, 140, 222, 264, 181, 243, 203, 238, 262, 362, 249, 183, 238, 226, 243, 294, 375, 455, 292, 245, 414

Offset: 1

N. J. A. Sloane, Apr 06 2001; b-file added Nov 27 2007

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..88 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Tables (gives many more terms)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A060410, A060411.

A060411 In the '3x+1' problem, take the sequence of starting values which set new records for the highest point of the trajectory before reaching 1 (A006884); sequence gives iterate where maximal value is reached in the trajectory with that start.

Original entry on oeis.org

0, 2, 3, 4, 45, 8, 31, 14, 48, 28, 49, 33, 26, 50, 35, 43, 67, 39, 69, 39, 35, 124, 83, 37, 132, 46, 81, 75, 83, 118, 68, 42, 72, 97, 98, 75, 92, 199, 109, 92, 92, 160, 91, 197, 119, 113, 109, 124, 176, 217, 234, 276, 172, 164, 233, 101, 109, 158, 157

Offset: 1

N. J. A. Sloane, Apr 06 2001; b-file added Nov 27 2007

nonn

N. J. A. Sloane, Table of n, a(n) for n = 1..88 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Tables (gives many more terms)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A060409, A060410.

A132348 Lengths of the 3x+1 trajectories associated with the record values in A006884.

Original entry on oeis.org

0, 1, 7, 16, 17, 111, 47, 97, 131, 170, 161, 201, 170, 184, 255, 307, 160, 334, 231, 247, 162, 183, 406, 441, 242, 439, 328, 367, 370, 427, 430, 399, 363, 322, 573, 400, 483, 576, 606, 483, 572, 438, 475, 592, 770, 726, 543, 836, 555, 770, 871, 796, 1109, 755

Offset: 1

N. J. A. Sloane, Nov 10 2007

nonn

R. B. Banks, Slicing Pizzas, Racing Turtles and Further Adventures in Applied Mathematics, Princeton Univ. Press, 1999. See p. 96.

Hugo Pfoertner, Table of n, a(n) for n = 1..97
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.

Cf. A006577, A006884.

a(n) = A006577(A006884(n)). - Hugo Pfoertner, Jan 13 2025

Extended by T. D. Noe, Apr 27 2010

A365482 In the Collatz (3x+1) problem, values in A006884 for which the maximum excursion ratio (see comments) is greater than 2.

Original entry on oeis.org

27, 319804831, 1410123943, 3716509988199, 9016346070511, 1254251874774375, 10709980568908647, 1980976057694848447

Offset: 1

Paolo Xausa, Sep 05 2023

Kontorovich and Lagarias (2009, 2010) define the maximum excursion ratio as the ratio between the log of the highest point in the trajectory of the T function (started at x) and the log of x, where T(x) is the 3x+1 function = (3x+1)/2 if x is odd, x/2 if x is even (A014682).
They use data from Oliveira e Silva (2010) to compile Table 3 in their paper, but they omit the a(7) = 10709980568908647 value (cf. also Barina and Roosendall links).
Equivalently, values in A006884 for which A365478(A006884(k)) / A006884(k)^2 > 1, for k >= 1.
See A365483 for corresponding maximum excursion values.

David Barina, Path records.
Alex V. Kontorovich and Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems, arXiv:0910.1944 [math.NT], 2009, pp. 11-14, and in Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 140-142.
Tomás Oliveira e Silva, Empirical Verification of the 3x+1 and Related Conjectures, in Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 189-207.
Eric Roosendall, 3x + 1 Path Records.
Index entries for sequences related to 3x+1 (or Collatz) problem

Subsequence of A006884.

Cf. A014682, A365478, A365483.

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

A006877 In the '3x+1' problem, these values for the starting value set new records for number of steps to reach 1.

Original entry on oeis.org

1, 2, 3, 6, 7, 9, 18, 25, 27, 54, 73, 97, 129, 171, 231, 313, 327, 649, 703, 871, 1161, 2223, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799

Offset: 1

N. J. A. Sloane and Robert Munafo

Both the 3x+1 steps and the halving steps are counted.

This sequence without a(2) = 2 specifies where records occur in A208981. - Omar E. Pol, Apr 14 2022

D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..148 (from Eric Rosendaal's 3x+1 Delay Records, terms 1..130 from T. D. Noe)
T. Ahmed and H. Snevily, Are there an infinite number of Collatz integers?, 2013.
David Barina, Convergence verification of the Collatz problem, The Journal of Supercomputing 77(3) (2021), 2681-2688.
David Barina, Computational Verification of the Collatz Problem, preprint on Research Square (2024).
Gaston H. Gonnet, Computations on the 3n+1 conjecture, Maple Technical Newsletter 6 (1991): 18-22.
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Robert Munafo, Integer Sequences Related to 3x+1 Collatz Iteration
Eric Roosendaal, 3x+1 Delay Records
Olivier Rozier and Claude Terracol, Paradoxical behavior in Collatz sequences, arXiv:2502.00948 [math.GM], 2025. See p. 15.
Robert G. Wilson v, Letter to N. J. A. Sloane with attachments, Jan. 1989
Robert G. Wilson v, Tables of A6877, A6884, A6885, Jan. 1989
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A006885, A006877, A006878, A033492, A380138.

A006877 := proc(n) local a,L; L := 0; a := n; while a <> 1 do if a mod 2 = 0 then a := a/2; else a := 3*a+1; fi; L := L+1; od: RETURN(L); end;

numberOfSteps[x0_] := Block[{x = x0, nos = 0}, While [x != 1 , If[Mod[x, 2] == 0 , x = x/2, x = 3*x + 1]; nos++]; nos]; a[1] = 1; a[n_] := a[n] = Block[{x = a[n-1] + 1}, record = numberOfSteps[x - 1]; While[ numberOfSteps[x] <= record, x++]; x]; A006877 = Table[ Print[a[n]]; a[n], {n, 1, 44}](* Jean-François Alcover, Feb 14 2012 *)
DeleteDuplicates[Table[{n,Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]},{n,838000}],GreaterEqual[#1[[2]],#2[[2]]]&][[All,1]] (* Harvey P. Dale, May 13 2022 *)

A006577(n)=my(s);while(n>1,n=if(n%2,3*n+1,n/2);s++);s
step(n,r)=my(t);forstep(k=bitor(n,1),2*n,2,t=A006577(k);if(t>r,return([k,t])));[2*n,r+1]
r=0;print1(n=1);for(i=1,100,[n,r]=step(n,r); print1(", "n)) \\ Charles R Greathouse IV, Apr 01 2013

c1 = lambda x: (3*x+1 if (x%2) else x>>1)
r = -1
for n in range(1, 10**5):
    a=0 ; n1=n
    while n>1: n=c1(n); a+=1;
    if a > r: print(n1, end = ', '); r=a
print('...') # Ya-Ping Lu and Robert Munafo, Mar 22 2024

A006878 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A006877.

Original entry on oeis.org

0, 1, 7, 8, 16, 19, 20, 23, 111, 112, 115, 118, 121, 124, 127, 130, 143, 144, 170, 178, 181, 182, 208, 216, 237, 261, 267, 275, 278, 281, 307, 310, 323, 339, 350, 353, 374, 382, 385, 442, 448, 469, 508, 524, 527, 530, 556, 559, 562, 583, 596, 612, 664, 685, 688, 691, 704

Offset: 1

N. J. A. Sloane, Robert Munafo

Both the 3x+1 steps and the halving steps are counted.

D. R. Hofstadter, Goedel, Escher, Bach: an Eternal Golden Braid, Random House, 1980, p. 400.
G. T. Leavens and M. Vermeulen, 3x+1 search problems, Computers and Mathematics with Applications, 24 (1992), 79-99.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Hugo Pfoertner, Table of n, a(n) for n = 1..148 (from Eric Rosendaal's 3x+1 Delay Records, terms 1..130 from T. D. Noe)
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Eric Roosendaal, 3x+1 Delay Records
Robert G. Wilson v, Letter to N. J. A. Sloane with attachments, Jan. 1989
Index entries for sequences from "Goedel, Escher, Bach"
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006884, A006885, A006877, A033492, A033958, A033959.

f := proc(n) local a,L; L := 0; a := n; while a <> 1 do if a mod 2 = 0 then a := a/2; else a := 3*a+1; fi; L := L+1; od: RETURN(L); end;

numberOfSteps[x0_] := Block[{x = x0, nos = 0}, While[x != 1, If[Mod[x, 2] == 0, x = x/2, x = 3*x+1]; nos++]; nos]; A006878 = numberOfSteps /@ A006877 (* Jean-François Alcover, Feb 22 2012 *)
DeleteDuplicates[Table[Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]],{n,0,10^6}],GreaterEqual]-1 (* The program generates the first 44 terms of the sequence, derived from all starting values from 1 up to and including 1 million. *) (* Harvey P. Dale, Nov 26 2022 *)

A033492 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A006877 (same as A006878 except here we start counting at 1 instead of 0).

Original entry on oeis.org

1, 2, 8, 9, 17, 20, 21, 24, 112, 113, 116, 119, 122, 125, 128, 131, 144, 145, 171, 179, 182, 183, 209, 217, 238, 262, 268, 276, 279, 282, 308, 311, 324, 340, 351, 354, 375, 383, 386, 443, 449, 470, 509, 525, 528, 531, 557, 560, 563, 584, 597, 613, 665, 686

Offset: 1

Jeff Burch

nonn

Both the 3x+1 steps and the halving steps are counted.

R. E. Maeder, Programming in Mathematica, 3rd Edition, Addison-Wesley, pages 251-252.

T. D. Noe, Table of n, a(n) for n=1..130 (from Eric Roosendaal's data)
Eric Roosendaal, 3x+1 Delay Records
Index entries for sequences related to 3x+1 (or Collatz) problem

Equal to A006878 + 1. Cf. A006884, A006885, A033492.

Corrected and extended by Lee Corbin (lcorbin(AT)tsoft.com)

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

cp's OEIS Frontend

A006885 Record highest point of trajectory before reaching 1 in '3x+1' problem, corresponding to starting values in A006884.

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A060410 In the '3x+1' problem, take the sequence of starting values which set new records for the highest point of the trajectory before reaching 1 (A006884); sequence gives associated maximal value reached in the trajectory with that start.

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A060409 In the '3x+1' problem, take the sequence of starting values which set new records for the highest point of the trajectory before reaching 1 (A006884); sequence gives associated "dropping time", number of steps to reach a lower value than the start.

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A060411 In the '3x+1' problem, take the sequence of starting values which set new records for the highest point of the trajectory before reaching 1 (A006884); sequence gives iterate where maximal value is reached in the trajectory with that start.

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A132348 Lengths of the 3x+1 trajectories associated with the record values in A006884.

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A365482 In the Collatz (3x+1) problem, values in A006884 for which the maximum excursion ratio (see comments) is greater than 2.

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

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A006877 In the '3x+1' problem, these values for the starting value set new records for number of steps to reach 1.

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Maple

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A006878 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A006877.

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