A033492 -id:A033492 - cp's OEIS frontend

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

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

A006884 In the '3x+1' problem, these values for the starting value set new records for highest point of trajectory before reaching 1.

Original entry on oeis.org

1, 2, 3, 7, 15, 27, 255, 447, 639, 703, 1819, 4255, 4591, 9663, 20895, 26623, 31911, 60975, 77671, 113383, 138367, 159487, 270271, 665215, 704511, 1042431, 1212415, 1441407, 1875711, 1988859, 2643183, 2684647, 3041127, 3873535, 4637979, 5656191

Offset: 1

N. J. A. Sloane, Robert Munafo

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

Where records occur in A025586: A006885(n) = A025586(a(n)) and A025586(m) < A006885(n) for m < a(n). - Reinhard Zumkeller, May 11 2013

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.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

David Barina, Table of n, a(n) for n = 1..98 (terms 1..84 from T. D. Noe, terms 85..89 from N. J. A. Sloane).
David Barina, Path records.
David Barina, Improved verification limit for the convergence of the Collatz conjecture, J. Supercomputing (2025) Vol. 81, Art. No. 810. See p. 12.
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 problems, Computers and Mathematics with Applications, 24 (1992), 79-99.
G. T. Leavens and M. Vermeulen, 3x+1 search programs, Computers and Mathematics with Applications, 24 (1992), 79-99. (Annotated scanned copy)
Tomás Oliveira e Silva, Tables (gives many more terms).
Eric Roosendaal, 3x+1 Path 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 related to 3x+1 (or Collatz) problem
Index entries for sequences from "Goedel, Escher, Bach"

A060409 gives associated "dropping times", A060410 the maximal values and A060411 the steps at which the maxima occur.

Cf. A006885, A006877, A006878, A033492, A060412-A060415, A132348.

a006884 n = a006884_list !! (n-1)
a006884_list = f 1 0 a025586_list where
   f i r (x:xs) = if x > r then i : f (i + 1) x xs else f (i + 1) r xs
-- Reinhard Zumkeller, May 11 2013

mcoll[n_]:=Max@@NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; t={1,max=2}; Do[If[(y=mcoll[n])>max,max=y; AppendTo[t,n]],{n,3,705000,4}]; t (* Jayanta Basu, May 28 2013 *)
DeleteDuplicates[Parallelize[Table[{n,Max[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]},{n,57*10^5}]],GreaterEqual[#1[[2]],#2[[2]]]&][[;;,1]] (* Harvey P. Dale, Apr 23 2023 *)

A025586(n)=my(r=n); while(n>2, if(n%2, n=3*n+1; if(n>r, r=n)); n>>=1); r
r=0; for(n=1,1e6, t=A025586(n); if(t>r, r=t; print1(n", "))) \\ Charles R Greathouse IV, May 25 2016

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

Original entry on oeis.org

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 *)

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 *)

A033958 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, 3, 7, 9, 25, 27, 73, 97, 129, 171, 231, 313, 327, 703, 871, 1161, 2463, 2919, 3711, 6171, 10971, 13255, 17647, 23529, 26623, 34239, 35655, 52527, 77031, 106239, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, 1117065, 1501353, 1723519, 2298025, 3064033

Offset: 1

N. J. A. Sloane

Only the 3x+1 steps, not 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.

Charles R Greathouse IV, Table of n, a(n) for n = 1..71
Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
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, A033959.

a033958 n = a033958_list !! (n-1)
-- For definition of a033958_list: see A033959.
-- Reinhard Zumkeller, Jan 08 2014

f[ nn_ ] := Module[ {c, n}, c = 0; n = nn; While[ n != 1, If[ Mod[ n, 2 ] == 0, n /= 2, n = 3*n + 1; c++ ] ]; Return[ c ] ] maxx = -1; For[ n = 1, n <= 10^8, n++, Module[ {val}, val = f[ n ]; If[ val > maxx, maxx = val; Print[ n, " ", val ] ] ] ] (* Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000 *)

Positions of records in A006667. - Sean A. Irvine, Jul 22 2020

More terms from Jud McCranie, Jan 26 2000
Corrected with Mathematica code by Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000
a(40)-a(43) from Charles R Greathouse IV, Oct 07 2013

A033959 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A033958.

Original entry on oeis.org

0, 2, 5, 6, 7, 41, 42, 43, 44, 45, 46, 47, 52, 62, 65, 66, 76, 79, 87, 96, 98, 101, 102, 103, 113, 114, 119, 125, 129, 130, 138, 141, 142, 164, 166, 174, 189, 195, 196, 197, 207, 208, 209, 217, 222, 228, 248, 256, 257, 258, 263, 278, 357, 358, 359, 362, 370

Offset: 1

N. J. A. Sloane

Only the 3x+1 steps, not 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.

Brian Hayes, Computer Recreations: On the ups and downs of hailstone numbers, Scientific American, 250 (No. 1, 1984), pp. 10-16.
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, A033958.

Cf. A075680.

a033959 n = a033959_list !! (n-1)
(a033959_list, a033958_list) = unzip $ (0, 1) : f 1 1 where
   f i x | y > x     = (y, 2 * i - 1) : f (i + 1) y
         | otherwise = f (i + 1) x
         where y = a075680 i
-- Reinhard Zumkeller, Jan 08 2014

A033959 := 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; L := L+1; fi; od: RETURN(L); end;

f[ nn_ ] := Module[ {c, n}, c = 0; n = nn; While[ n != 1, If[ Mod[ n, 2 ] == 0, n /= 2, n = 3*n + 1; c++ ] ]; Return[ c ] ] maxx = -1; For[ n = 1, n <= 10^8, n++, Module[ {val}, val = f[ n ]; If[ val > maxx, maxx = val; Print[ n, " ", val ] ] ] ]

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 27 2000
More terms from Larry Reeves (larryr(AT)acm.org), Sep 27 2000
Offset corrected by Reinhard Zumkeller, Jan 08 2014

A143812 Maximal number of halving and tripling steps to reach 1 in '3x+1' problem for range (1, ..., n).

Original entry on oeis.org

1, 2, 8, 8, 8, 9, 17, 17, 20, 20, 20, 20, 20, 20, 20, 20, 20, 21, 21, 21, 21, 21, 21, 21, 24, 24, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 112, 113, 113, 113, 113, 113

Offset: 1

Benjamin Frost (benjamin.frost(AT)students.adelaide.edu.au), Sep 02 2008

nonn

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006877, A033492.

nst[n_]:=Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]; nn=60; With[ {stps= Array[nst,nn]},Table[Max[Take[stps,n]],{n,nn}]]  (* Harvey P. Dale, Apr 17 2014 *)

Corrected and extended by Harvey P. Dale, Apr 17 2014

A339614 Inputs n that yield a record-breaking value of A008908(n)/(log_2(n)+1) for the Collatz conjecture.

Original entry on oeis.org

1, 3, 7, 9, 27, 26623, 35655, 52527, 142587, 156159, 230631, 626331, 837799, 1723519, 3542887, 3732423, 5649499, 6649279, 8400511, 63728127, 3743559068799, 100759293214567, 104899295810901231

Offset: 1

Matthew Russell Downey, Dec 10 2020

The metric A008908(n)/(log_2(n)+1) is always equal to 1 for any power of 2 (where 1 is the smallest possible value).

			a(1) = 1, which is trivial, because the first element in any sequence is record setting.
a(5) = 27, because A008908(n)/(log_2(n)+1) yields a maximum value at n=27 among the first 27 elements, and there are 4 record-breaking elements beforehand.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A008908, A006877, A033492.

import math
oeis_A006877 = [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, 10623, 142587, 156159, 216367, 230631, 410011, 511935, 626331, 837799, 1117065, 1501353, 1723519, 2298025, 3064033, 3542887, 3732423, 5649499, 6649279, 8400511, 11200681, 14934241, 15733191, 31466382, 36791535, 63728127]
def stopping_time(n):
    time = 1
    while n>1:
        n = 3*n + 1 if n & 1 else n//2
        time += 1
    return time
def stopping_time_metric(n):
    time = stopping_time(n)
    logarithmic_distance = (math.log(n, 2)+1)
    return float(time/logarithmic_distance)
result = []
record_input = oeis_A006877[0]
record_stopping_time_metric = stopping_time_metric(record_input)
result.append(record_input)
for n in range(1, len(oeis_A006877)):
    current_input = oeis_A006877[n]
    current_stopping_time_metric = stopping_time_metric(current_input)
    if current_stopping_time_metric > record_stopping_time_metric:
        record_input = current_input
        record_stopping_time_metric = current_stopping_time_metric
        result.append(record_input)
for n in range(len(result)):
    print(result[n], end=", ")

cp's OEIS Frontend

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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A006884 In the '3x+1' problem, these values for the starting value set new records for highest point of trajectory before reaching 1.

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

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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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A033958 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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A033959 Record number of steps to reach 1 in '3x+1' problem, corresponding to starting values in A033958.

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A143812 Maximal number of halving and tripling steps to reach 1 in '3x+1' problem for range (1, ..., n).

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