A006878 -id:A006878 - cp's OEIS frontend

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).

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

A288493 First differences of A006878 (record new trajectory lengths of Collatz function) (Hailstone sequence).

Original entry on oeis.org

1, 6, 1, 8, 3, 1, 3, 88, 1, 3, 3, 3, 3, 3, 3, 13, 1, 26, 8, 3, 1, 26, 8, 21, 24, 6, 8, 3, 3, 26, 3, 13, 16, 11, 3, 21, 8, 3, 57, 6, 21, 39, 16, 3, 3, 26, 3, 3, 21, 13, 16, 52, 21, 3, 3, 13, 1, 39, 205, 1, 3, 3, 8, 1, 21, 1, 13, 8, 42, 37, 44, 1, 21, 31, 26, 3, 6, 1, 8, 6, 8, 13, 52, 1, 13, 3, 8, 3, 13, 8, 52, 3, 26, 3, 3, 106, 1, 13, 3, 3, 16, 3, 13, 16, 21, 13, 8

Offset: 1

David Rabahy, Jun 13 2017

nonn

The sequence appears to return to 1 again and again forever when the minimal possible new record of just (previous record + 1) is reached at the latest possible value of 2X. Through 129 there are only 2 entries, a(17) and a(21), that are a minimal new record but aren't 2X.

			For n = 3 the difference between A006878(4) = 8 and A006878(3) = 7 is 1.

Hugo Pfoertner, Table of n, a(n) for n = 1..147 (from Eric Rosendaal's 3x+1 Delay Records, terms 1..129 from David Rabahy)
Eric Roosendaal, 3x+1 Delay Records

Cf. A006877, A006878, A375094.

(* This script is not suitable to compute a large number of terms. *)
terms = 40; steps[x0_] := steps[x0] = Block[{x = x0, nos = 0}, While[x != 1, If[Mod[x, 2] == 0, x = x/2, x = 3*x + 1]; nos++]; nos]; b[1] = 1; b[n_] := b[n] = Block[{x = b[n - 1] + 1}, record = steps[x - 1]; While[steps[x] <= record, x++]; x];
A006877 = Table[Print[b[n]]; b[n], {n, 1, terms+1}];
A006878 = steps /@ A006877;
Differences[A006878] (* Jean-François Alcover, Jun 15 2017 *)

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

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 *)

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

A171938 Record values in A138753 (a "prime" variation of the Collatz (3n+1) problem).

Original entry on oeis.org

1, 4, 5, 8, 21, 22, 24, 29, 60, 61, 72, 73, 97, 100, 184, 216, 239, 451, 469

Offset: 1

M. F. Hasler, Apr 01 2008

Cf. A124123, A138750-A138754, A138756, A006878 (analog for Collatz problem).

A138754[n_]:=A138754[n]=With[{p=Prime[n]},PrimePi[NextPrime[If[Mod[p,3]==2,p/2,2p]]]];
A138753[n_]:=Length[NestWhileList[A138754,n,UnsameQ,{1,4}]]-1;
A171938list[upto_]:=Module[{v,r=0},Table[If[(v=A138753[n])>r,r=v,Nothing],{n,upto}]];
A171938list[500] (* Paolo Xausa, Jul 29 2023 *)

m=0; for( i=1,#A138753, A138753[i] > m & print1( m=A138753[i],", "))

A171938(n) = A138753(A138756(n)).

A171938 = { A138753(m) | A138753(k) < A138753(m) for all k

Originally submitted as A138755, but mislaid by Editor-in-Chief; renumbered and added to OEIS, Oct 24 2010

a(15)-a(19) from Paolo Xausa, Jul 29 2023

A125195 Records in A127885.

Original entry on oeis.org

0, 1, 7, 8, 16, 17, 20, 23, 31, 32, 38, 46, 47, 50, 53, 56, 59

Offset: 1

David Applegate and N. J. A. Sloane, Feb 06 2007

Cf. A127885, A125686, A006878.

A380138 a(n) is the largest value in the '3x+1' trajectory of starting points producing a record number of steps.

Original entry on oeis.org

1, 2, 16, 16, 52, 52, 52, 88, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 250504, 190996, 190996, 250504, 250504, 250504, 481624, 975400, 975400, 497176, 11003416, 11003416, 106358020, 18976192, 41163712, 106358020, 21933016, 104674192, 593279152

Offset: 1

Hugo Pfoertner, Jan 13 2025

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..148

Cf. A006877, A006878, A006884, A006885, A025586.

s = Map[ToExpression,
  StringSplit[
    Import["https://oeis.org/A006877/b006877.txt", "Data"][[2 ;; -1]]
  ][[All, -1]] ];
Map[Max@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, #, # > 1 &] &, s] (* Michael De Vlieger, Jan 13 2025 *)

a(n) = A025586(A006877(n)).

Showing 1-10 of 10 results.

cp's OEIS Frontend

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).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A288493 First differences of A006878 (record new trajectory lengths of Collatz function) (Hailstone sequence).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

A033958 In the '3x+1' problem, these values for the starting value set new records for number of steps to reach 1.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple