A025586 -id:A025586 - 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

A033496 Numbers m that are the largest number in their Collatz (3x+1) trajectory.

Original entry on oeis.org

1, 2, 4, 8, 16, 20, 24, 32, 40, 48, 52, 56, 64, 68, 72, 80, 84, 88, 96, 100, 104, 112, 116, 128, 132, 136, 144, 148, 152, 160, 168, 176, 180, 184, 192, 196, 200, 208, 212, 224, 228, 232, 240, 244, 256, 260, 264, 272, 276, 280, 288, 296, 304, 308, 312, 320, 324

Offset: 1

Jeff Burch

Or, possible peak values in 3x+1 trajectories: 1,2 and m=16k+4,16k+8,16k but not for all k; those 4k numbers [like m=16k+12 and others] which cannot be such peaks are listed in A087252.

Possible values of A025586(m) in increasing order. See A275109 (number of times each value of a(n) occurs in A025586). - Jaroslav Krizek, Jul 17 2016

			These peak values occur in 1, 3, 6, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, 21, 22, 27, 30, 39, 44, 71, 75, 1579 [3x+1]-iteration trajectories started with different initial values. This list most probably is incomplete.
From _Hartmut F. W. Hoft_, Jun 24 2016: (Start)
Let n be the maximum in some Collatz trajectory and let F(n), the initial fan of n, be the set of all initial values less than or equal to n whose Collatz trajectories lead to n as their maximum. Then the size of F(n) never equals 2, 4, 5, 7 or 10 (see the link).
Conjecture: Every number k > 10 occurs as the size of F(n) for some n.
Fans F(n) of size k, for all 10 < k < 355, exist for 4 <= n <= 50,000,000. The largest fan in this range, F(41163712), has size 7450.
(End)

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000 (first 2000 terms from T. D. Noe)
Hartmut F. W. Hoft, initial Collatz fans
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A025586, A087251-A087256, A015999, A275109.
Cf. A006370, A008908, A070165.
Cf. A095384 (contains a definition of Collatz[]).
Cf. A105730, A233293.

a033496 n = a033496_list !! (n-1)
a033496_list = 1 : filter f [2, 4 ..] where
   f x = x == maximum (takeWhile (/= 1) $ iterate a006370 x)
-- Reinhard Zumkeller, Oct 22 2015

Set(Sort([Max([k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]): n in [1..2^10] | Max([k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]) le 2^10])) // Jaroslav Krizek, Jul 17 2016

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 *)
Select[Range[324], Max[Collatz[#]] == # &] (* T. D. Noe, Feb 28 2013 *)

def a(n):
    if n<2: return [1]
    l=[n, ]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        if n not in l:
            l.append(n)
            if n<2: break
        else: break
    return l
print([n for n in range(1, 501) if max(a(n)) == n]) # Indranil Ghosh, Apr 14 2017

A008908(a(n)) = A159999(a(n)). - Reinhard Zumkeller, May 04 2009

Max(A070165(a(n),k): k=1..A008908(a(n))) = A070165(a(n),1) = a(n). - Reinhard Zumkeller, Oct 22 2015

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 *)

A056959 In repeated iterations of function m -> m/2 if m even, m -> 3m+1 if m odd, a(n) is maximum value achieved if starting from n.

Original entry on oeis.org

4, 4, 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

Offset: 1

Henry Bottomley, Jul 18 2000

nonn

If a(n) exists (which is the essence of the "3x+1" problem) then a(n) must be a multiple of 4, since if a(n) was odd then the next iteration 3*a(n)+1 would be greater than a(n), while if a(n) was twice an odd number then the next-but-one iteration (3/2)*a(n)+1 would be greater.

The variant A025586 considers the trajectory ending in 1, by definition. Therefore the two sequences differ just at a(1) and a(2). - M. F. Hasler, Oct 20 2019

			a(6) = 16 since iteration starts: 6, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, ... and 16 is highest value.

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

Cf. A006370, A056957, A056958.

Essentially the same as A025586.

a:= proc(n) option remember; `if`(n=1, 4,
      max(n, a(`if`(n::even, n/2, 3*n+1))))
    end:
seq(a(n), n=1..88);  # Alois P. Heinz, Oct 16 2021

a[n_] := Module[{r = n, m = n}, If[n <= 2, 4, While[m > 2, If[OddQ[m], m = 3*m + 1; If[m > r, r = m], m = m/2]]; r]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, May 20 2022 *)

a(n)=my(r=max(4,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

A220237 Triangle read by rows: sorted terms of Collatz trajectories.

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Jan 03 2013

n-th row = sorted list of {A070165(n,k): k = 1..A006577(n)};

T(n,1) = 1 if Collatz conjecture is true.

			The table begins:
.   1:  [1]
.   2:  [1,2]
.   3:  [1,2,3,4,5,8,10,16]
.   4:  [1,2,4]
.   5:  [1,2,4,5,8,16]
.   6:  [1,2,3,4,5,6,8,10,16]
.   7:  [1,2,4,5,7,8,10,11,13,16,17,20,22,26,34,40,52]
.   8:  [1,2,4,8]
.   9:  [1,2,4,5,7,8,9,10,11,13,14,16,17,20,22,26,28,34,40,52]
.  10:  [1,2,4,5,8,10,16]
.  11:  [1,2,4,5,8,10,11,13,16,17,20,26,34,40,52]
.  12:  [1,2,3,4,5,6,8,10,12,16] .

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577 (row lengths), A025586(right edge), A033493 (row sums).

import Data.List (sort)
a220237 n k = a220237_tabf !! (n-1) !! (k-1)
a220237_row n = a220237_tabf !! (n-1)
a220237_tabf = map sort a070165_tabf

T:= proc(n) option remember; `if`(n=1, 1,
      sort([n, T(`if`(n::even, n/2, 3*n+1))])[])
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Oct 16 2021

Flatten[Table[Sort[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]],{n,12}]] (* Harvey P. Dale, Jan 28 2013 *)

A087256 Number of different initial values for 3x+1 trajectories in which the largest term appearing in the iteration is 2^n.

Original entry on oeis.org

1, 1, 1, 6, 1, 3, 1, 3, 1, 12, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 13, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 9, 1, 3, 1, 3, 1, 11, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 21, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 78, 1, 3, 1, 3, 1, 8, 1, 3, 1, 3, 1, 6, 1, 3, 1, 3, 1, 9, 1, 3, 1

Offset: 1

Labos Elemer, Sep 08 2003

nonn

It would be interesting to know whether the ...1,3,1,3,1,x,1,3,1,3,1,... pattern persists. - John W. Layman, Jun 09 2004
The observed pattern should persist. Proof: [1] a(odd)=1 because -1+2^odd is not divisible by 3, so in Collatz-algorithm 2^odd is preceded by increasing inverse step. Thus 2^odd is the only suitable initial value; [2] a[2k]>=3 for k>1 because 2^(2k)-1=-1+4^k=3A so {b=2^2k, (b-1)/3 and (2a-2)/3} are three relevant initial values. No more case arises unless condition-[3] (see below) was satisfied; [3] a[6k+4]>=5 for k>=1, ..iv=c=2^(6k+4); here {c, (c-1)/3, 2(c-1)/3, (2c-5)/9, (4c-10)/9} is 5 suitable initial values, iff (2c-5)/9 is an integer; e.g. at 6k+4=10, {1024<-341<-682<-227<-454} back-tracking the iteration. - Labos Elemer, Jun 17 2004
From Hartmut F. W. Hoft, Jun 24 2016: (Start)
Except for a(2)=1 the sequence has the 6-element quasiperiod 1, 3, 1, x, 1, 3  where x>=6, but unequal to 7 and 10 (see links below and in A033496). Observe that for n=2^(6k+4)=16*2^(6k), n mod 9 = 7 so that (2n-5)/9 is an integer and a(n)>=6.
Conjecture: All numbers m > 10 occur as values in A087256 (see A233293).
The conjecture has been verified for all 10 < k < 133 for Collatz trajectories with maximum value through 2^(36000*6 + 4). The largest fan of initial values in this range, F(6*1993+4), has maximum 2^11962 and size 3958.
(End)

			n = 10: 2^10 = 1024 = peak for trajectories started with initial value taken from the list: {151, 201, 227, 302, 341, 402, 454, 604, 682, 804, 908, 1024};
a trajectory with peak=1024: {201, 604, 302, 151, 454, 227, 682, 341, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1}

Hartmut F. W. Hoft, proof of quasi period 6

Cf. A025586, A087251, A087252, A087253, A087254, A105730, A233293.

c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=FixedPointList[c, x]; {$RecursionLimit=1000;m=0}; Table[Print[{xm-1, m}];m=0; Do[If[Equal[Max[fpl[n]], 2^xm], m=m+1], {n, 1, 2^xm}], {xm, 1, 30}]

f(n, m) = 1 + if(2*n <= m, f(2*n, m), 0) + if (n%6 == 4, f(n\3, m), 0);
a(n) = f(2^n, 2^n); \\ David Wasserman, Apr 18 2005

a(6n+4) = A105730(n). - David Wasserman, Apr 18 2005

Terms a(19)-a(21) from John W. Layman, Jun 09 2004

More terms from David Wasserman, Apr 18 2005

A087272 a(n) is the largest prime number in 3x+1 trajectory initiated at n.

Original entry on oeis.org

2, 5, 2, 5, 5, 17, 2, 17, 5, 17, 5, 13, 17, 53, 2, 17, 17, 29, 5, 2, 17, 53, 5, 29, 13, 1619, 17, 29, 53, 1619, 2, 29, 17, 53, 17, 37, 29, 101, 5, 1619, 2, 43, 17, 17, 53, 1619, 5, 37, 29, 29, 13, 53, 1619, 1619, 17, 43, 29, 101, 53, 61, 1619, 1619, 2, 37, 29, 101, 17, 13, 53

Offset: 2

Labos Elemer, Sep 18 2003

nonn

Rémy Sigrist, Table of n, a(n) for n = 2..10000

Cf. A025586, A055510, A087232.

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] ofp[x_] := Part[fpl[x], Flatten[Position[PrimeQ[fpl[x]], True]]] Table[Max[ofp[w]], {w, 1, 256}]
Table[Max[Select[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],PrimeQ]],{n,2,70}] (* Harvey P. Dale, Feb 27 2023 *)

a(n) = my (mx=2); while (n>1, if (isprime(n), mx=max(mx,n)); n=if (n%2, 3*n+1, n/2)); mx \\ Rémy Sigrist, Oct 08 2018

Offset corrected by Rémy Sigrist, Oct 08 2018

A087232 a(n) is the largest odd term in the 3x+1 trajectory initiated at n.

Original entry on oeis.org

1, 1, 5, 1, 5, 5, 17, 1, 17, 5, 17, 5, 13, 17, 53, 1, 17, 17, 29, 5, 21, 17, 53, 5, 29, 13, 3077, 17, 29, 53, 3077, 1, 33, 17, 53, 17, 37, 29, 101, 5, 3077, 21, 65, 17, 45, 53, 3077, 5, 49, 29, 77, 13, 53, 3077, 3077, 17, 65, 29, 101, 53, 61, 3077, 3077, 1, 65, 33, 101, 17, 69

Offset: 1

Labos Elemer, Sep 18 2003

nonn

a(n)=3077 corresponds to peak=9232.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A025586, A056959, A348006.

a:= proc(n) option remember; `if`(n=1, 1, max(
     `if`(n::odd, n, 0), a(`if`(n::even, n/2, 3*n+1))))
    end:
seq(a(n), n=1..88);  # Alois P. Heinz, Nov 14 2021

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] ofp[x_] := Part[fpl[x], Flatten[Position[OddQ[fpl[x]], True]]] Table[Max[ofp[w]], {w, 1, 256}]
(* Second program: *)
Array[Max@ Select[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, Unequal[#, 1, -1, -10, -34] &, 1, 10^4], OddQ] &, 69] (* Michael De Vlieger, May 15 2017, after Alonso del Arte at A025586 *)

If n = 2^k (for integers k >= 0), a(n) = 1; otherwise a(n) = (A025586(n)-1)/3 =(A056959(n)-1)/3. - Paolo Xausa, Nov 13 2021

Name simplified by Paolo Xausa, Nov 13 2021

A095381 Initial values for 3x+1 trajectories in which the largest term arising in the iteration is a power of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 16, 21, 32, 42, 64, 85, 128, 151, 170, 201, 227, 256, 302, 341, 402, 454, 512, 604, 682, 804, 908, 1024, 1365, 2048, 2730, 4096, 5461, 8192, 10922, 14563, 16384, 19417, 21845, 29126, 32768, 38834, 43690, 58252, 65536, 87381

Offset: 1

Labos Elemer, Jun 14 2004

nonn

Clearly the sequence is infinite and a(n) < 2^n. - Charles R Greathouse IV, May 25 2016

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A025586, A087256, A209229, A225124.

// Valid below A006884(47) = 12327829503 on 64-bit machines.
static long is (unsigned long n) {
  unsigned long r = n;
  n >>= __builtin_ctzl(n); // gcc builtin for A007814
  while (n > 1) {
    n = 3*n + 1;
    if (n > r) r = n;
    n >>= __builtin_ctzl(n);
  }
  return !(r & (r-1));
} // Charles R Greathouse IV, May 25 2016

a095381 n = a095381_list !! (n-1)
a095381_list = map (+ 1) $ elemIndices 1 $ map a209229 a025586_list
-- Reinhard Zumkeller, Apr 30 2013

Coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3*#+1] &,n,#>1&];t={};Do[x = Max[Coll[n]];If[IntegerQ[Log[2,x]],AppendTo[t,n]],{n,90000}];t (* Jayanta Basu, Apr 28 2013 *)

is(n)=my(r=n); while(n>2, if(n%2, n=3*n+1; if(n>r, r=n)); n>>=1); r>>valuation(r,2)==1 \\ Charles R Greathouse IV, May 25 2016

A025586(a(n)) = 2^j for some j.

A233293 Smallest number that is the largest value in the Collatz (3x + 1) trajectories of exactly n initial values. (a(n)=0 if no such number exists.)

Original entry on oeis.org

3, 1, 0, 40, 0, 0, 16, 0, 88, 592, 0, 628, 52, 160, 304, 1672, 808, 2248, 3616, 11176, 10096, 8728, 4192, 23056, 13912, 65428, 40804, 5812, 9448, 12148, 8584, 82132, 27700, 10528, 91672, 53188, 58804, 20896, 96064, 2752, 32776, 25972, 14560, 183688, 8080

Offset: 0

Jon E. Schoenfield, Dec 06 2013

nonn

Smallest number that appears exactly n times in A025586.
Numbers that are not the largest value in the 3x + 1 trajectory of any initial value (that is, numbers that do not appear at all in A025586) are in A213199; the smallest such number is a(0) = 3.
Numbers that are the largest value in the 3x + 1 trajectory of exactly one initial value (that is, numbers that appear exactly once in A025586) are in A222562; the smallest such number is a(1) = 1.
Numbers that are the largest value in the 3x + 1 trajectories of exactly three initial values (that is, numbers that appear exactly three times in A025586) are in A232870; the smallest such number is a(3) = 40.
No number that is the largest value in the 3x + 1 trajectories of exactly 2, 4, 5, 7, or 10 initial values exists, so a(n) = 0 at n = 2, 4, 5, 7, and 10; for all other values of n up to 3000, a(n) > 0. Conjecture: a(n) > 0 for all n > 10. - Jon E. Schoenfield, Dec 14 2013

			a(0) = 3 because no 3x + 1 trajectories have 3 as their largest value, and 3 is the smallest number for which this is the case.
a(1) = 1 because exactly one 3x + 1 trajectory (namely, the one whose initial value is 1) has 1 as its largest value (and 1 is the smallest number for which this is the case).
a(3) = 40 because exactly three 3x + 1 trajectories (the ones whose initial values are 13, 26, and 40) have 40 as their largest value, and 40 is the smallest number for which this is the case.
a(2) = 0 because there exists no number that is the largest value in exactly two 3x + 1 trajectories.

Jon E. Schoenfield, Table of n, a(n) for n = 0..3000

Cf. A025586, A213199, A222562, A232870.

CollatzSeq[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 250000; c = Table[Max[CollatzSeq[n]], {n, nn}]; srt = Tally[Sort[c]]; times = Transpose[srt][[2]]; u = Union[times]; d = Differences[u]; mx = Position[d, ?(# > 1 &), 1, 1][[1, 1]] - 1; u2 = Transpose[Take[srt, 10]][[1]]; t0 = Complement[Range[u2[[-1]]], u2][[1]]; t = Table[k = srt[[Position[times, n, 1, 1][[1, 1]]]]; If[k[[1]] > nn, 0, k[[1]]], {n, mx}]; t = Join[{t0}, t] (* _T. D. Noe, Dec 07 2013 *)

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.

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A033496 Numbers m that are the largest number in their Collatz (3x+1) trajectory.

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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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A056959 In repeated iterations of function m -> m/2 if m even, m -> 3m+1 if m odd, a(n) is maximum value achieved if starting from n.

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A220237 Triangle read by rows: sorted terms of Collatz trajectories.

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A087256 Number of different initial values for 3x+1 trajectories in which the largest term appearing in the iteration is 2^n.

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A087272 a(n) is the largest prime number in 3x+1 trajectory initiated at n.

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