A060415 -id:A060415 - cp's OEIS frontend

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

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

A060412 In the '3x+1' problem, these values for the starting value set new records for the "dropping time", number of steps to reach a lower value than the start.

Original entry on oeis.org

2, 3, 7, 27, 703, 10087, 35655, 270271, 362343, 381727, 626331, 1027431, 1126015, 8088063, 13421671, 20638335, 26716671, 56924955, 63728127, 217740015, 1200991791, 1827397567, 2788008987, 12235060455

Offset: 1

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

nonn

The (3x+1)/2 steps and the halving steps are counted. - Don Reble, May 13 2006

Where records occur in A102419 (could be prefixed by an initial 1). - N. J. A. Sloane, Oct 20 2012

			See A102419.

N. J. A. Sloane, Table of n, a(n) for n = 1..35 (from the web page of Tomás Oliveira e Silva)
Tomás Oliveira e Silva, Tables
Eric Roosendaal, On the 3x + 1 problem
N. J. A. Sloane, First 36 terms of A217934 and A060412 [From Roosendaal web site]
Index entries for sequences related to 3x+1 (or Collatz) problem

A060413 gives associated "dropping times", A060414 the maximal values and A060415 the steps at which the maxima occur. See also A217934.

Cf. A060445, A008884, A161021, A161022, A161023, A014682, A126241.

dcoll[n_]:=Length[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>=n&]]; t={max=2}; Do[If[(y=dcoll[n])>max,max=y; AppendTo[t,n]],{n,3,1130000,4}]; t (* Jayanta Basu, May 28 2013 *)

A060445 "Dropping time" in 3x+1 problem starting at 2n+1 (number of steps to reach a lower number than starting value). Also called glide(2n+1).

Original entry on oeis.org

0, 6, 3, 11, 3, 8, 3, 11, 3, 6, 3, 8, 3, 96, 3, 91, 3, 6, 3, 13, 3, 8, 3, 88, 3, 6, 3, 8, 3, 11, 3, 88, 3, 6, 3, 83, 3, 8, 3, 13, 3, 6, 3, 8, 3, 73, 3, 13, 3, 6, 3, 68, 3, 8, 3, 50, 3, 6, 3, 8, 3, 13, 3, 24, 3, 6, 3, 11, 3, 8, 3, 11, 3, 6, 3, 8, 3, 65, 3, 34, 3, 6, 3, 47, 3, 8, 3, 13, 3, 6, 3, 8, 3

Offset: 0

N. J. A. Sloane, Apr 07 2001

If the starting value is even then of course the next step in the trajectory is smaller (cf. A102419).

The dropping time can be made arbitrarily large: If the starting value is of form n(2^m)-1 and m > 1, the next value is 3n(2^m)-3+1. That divided by 2 is 3n(2^(m-1))-1. It is bigger than the starting value and of the same form - substitute 3n -> n and m-1 -> m, so recursively get an increasing subsequence of m odd values. The dropping time is obviously longer than that. This holds even if Collatz conjecture were refuted. For example, m=5, n=3 -> 95, 286, 143, 430, 215, 646, 323, 970, 485, 1456, 728, 364, 182, 91. So the subsequence in reduced Collatz variant is 95, 143, 215, 323, 485. - Juhani Heino, Jul 21 2017

			3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2, taking 6 steps, so a(1) = 6.

T. D. Noe, Table of n, a(n) for n = 0..10000
Jason Holt, Plot of first 1 billion terms, log scale on x axis
Jason Holt, Plot of first 10 billion terms, log scale on x axis
Eric Roosendaal, On the 3x + 1 problem
Index entries for sequences related to 3x+1 (or Collatz) problem

A060565 gives the first lower number that is reached. Cf. A060412-A060415, A217934.
See A074473, A102419 for other versions of this sequence.
Cf. A122437 (allowable dropping times), A122442 (least k having dropping time A122437(n)).
Cf. A070165.

a060445 0 = 0
a060445 n = length $ takeWhile (>= n') $ a070165_row n'
            where n' = 2 * n + 1
-- Reinhard Zumkeller, Mar 11 2013

nxt[n_]:=If[OddQ[n],3n+1,n/2]; Join[{0},Table[Length[NestWhileList[nxt, n,#>=n&]]-1, {n,3,191,2}]]  (* Harvey P. Dale, Apr 23 2011 *)

def a(n):
    if n<1: return 0
    n=2*n + 1
    N=n
    x=0
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        x+=1
        if nIndranil Ghosh, Apr 22 2017

More terms from Jason Earls, Apr 08 2001 and from Michel ten Voorde Apr 09 2001

Still more terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2001

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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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A060412 In the '3x+1' problem, these values for the starting value set new records for the "dropping time", number of steps to reach a lower value than the start.

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A060445 "Dropping time" in 3x+1 problem starting at 2n+1 (number of steps to reach a lower number than starting value). Also called glide(2n+1).

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