A159999 -id:A159999 - cp's OEIS frontend

A008908 a(n) = (1 + number of halving and tripling steps to reach 1 in the Collatz (3x+1) problem), or -1 if 1 is never reached.

1, 2, 8, 3, 6, 9, 17, 4, 20, 7, 15, 10, 10, 18, 18, 5, 13, 21, 21, 8, 8, 16, 16, 11, 24, 11, 112, 19, 19, 19, 107, 6, 27, 14, 14, 22, 22, 22, 35, 9, 110, 9, 30, 17, 17, 17, 105, 12, 25, 25, 25, 12, 12, 113, 113, 20, 33, 20, 33, 20, 20, 108, 108, 7, 28, 28, 28, 15, 15, 15, 103

Offset: 1

N. J. A. Sloane, Bill Gosper

The number of steps (iterations of the map A006370) to reach 1 is given by A006577, this sequence counts 1 more. - M. F. Hasler, Nov 05 2017

When Collatz 3N+1 function is seen as an isometry over the dyadics, the halving step necessarily following each tripling is not counted, hence N -> N/2, if even, but N -> (3N+1)/2, if odd. Counting iterations of this map until reaching 1 leads to sequence A064433. - Michael Vielhaber (vielhaber(AT)gmail.com), Nov 18 2009

R. K. Guy, Unsolved Problems in Number Theory, E16.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
J. C. Lagarias, The 3x+1 problem and its generalizations, Amer. Math. Monthly, 92 (1985), 3-23.
Nitrxgen, Collatz Calculator
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577, A006370, A006667, A075677.

a008908 = length . a070165_row
-- Reinhard Zumkeller, May 11 2013, Aug 30 2012, Jul 19 2011

a:= proc(n) option remember; 1+`if`(n=1, 0,
      a(`if`(n::even, n/2, 3*n+1)))
    end:
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 29 2021

Table[Length[NestWhileList[If[EvenQ[ # ], #/2, 3 # + 1] &, i, # != 1 &]], {i, 75}]

a(n)=my(c=1); while(n>1, n=if(n%2, 3*n+1, n/2); c++); c \\ Charles R Greathouse IV, May 18 2015

def a(n):
    if n==1: return 1
    x=1
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        x+=1
        if n<2: break
    return x
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Apr 15 2017

a(n) = A006577(n) + 1.
a(n) = f(n,1) with f(n,x) = if n=1 then x else f(A006370(n),x+1).
a(A033496(n)) = A159999(A033496(n)). - Reinhard Zumkeller, May 04 2009
a(n) = A006666(n) + A078719(n).
a(n) = length of n-th row in A070165. - Reinhard Zumkeller, May 11 2013

More terms from Larry Reeves (larryr(AT)acm.org), Apr 27 2001
"Escape clause" added to definition by N. J. A. Sloane, Jun 06 2017
Edited by M. F. Hasler, Nov 05 2017

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

A076228 Number of terms k in the trajectory of the Collatz function applied to n such that k < n.

Original entry on oeis.org

0, 1, 2, 2, 3, 5, 4, 3, 6, 5, 6, 8, 6, 9, 6, 4, 8, 13, 10, 7, 5, 11, 8, 10, 13, 9, 9, 15, 13, 10, 9, 5, 16, 11, 8, 19, 17, 16, 17, 8, 12, 7, 19, 15, 13, 11, 12, 11, 19, 20, 17, 11, 9, 17, 14, 19, 23, 18, 21, 15, 13, 16, 14, 6, 22, 24, 21, 14, 12, 11, 15, 22, 18, 21, 7, 21, 19, 25, 22, 9

Offset: 1

Labos Elemer, Oct 01 2002

nonn

It is believed that for each x, a(n) = x occurs a finite number of times and the largest n is 2^x.

Original name: Start iteration of Collatz-function (A006370) with initial value of n. a(n) shows how many times during fixed-point-list, the value sinks below initial one until reaching endpoint = 1. - Michael De Vlieger, Dec 13 2018

			A070165(18) = {18, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1}. a(18) = 13 because 13 terms are smaller than n = 18; namely: {9, 14, 7, 11, 17, 13, 10, 5, 16, 8, 4, 2, 1}.

Michael De Vlieger, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070165, A074473, A159999.

f[x_] := (1-Mod[x, 2])*(x/2)+(Mod[x, 2])*(3*x+1) f[1]=1; f0[x_] := Delete[FixedPointList[f, x], -1] f1[x_] := f0[x]-Part[f0[x], 1] f2[x_] := Count[Sign[f1[x]], -1] Table[f2[w], {w, 1, 256}]
(* Second program: *)
Table[Count[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # > 1 &], ?(# < n &)], {n, 80}] (* _Michael De Vlieger, Dec 09 2018 *)

A160000 Number of partitions of n into parts occurring in '3x+1'-trajectory starting with n.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 9, 10, 17, 20, 23, 66, 35, 57, 51, 36, 77, 153, 122, 126, 61, 213, 189, 883, 353, 338, 337, 851, 645, 570, 571, 202, 1220, 1066, 901, 4017, 3462, 2440, 2777, 1598, 1852, 512, 8084, 4909, 3952, 3122, 3399, 29851, 17813, 11391, 11285, 7128

Offset: 1

Reinhard Zumkeller, May 11 2009

			7-22-11-34-17-52-26-13-40-20-10-5-16-8-[4-2-1]*:
{1,2,4,5,7} is the set of numbers <= n, occurring in this trajectory, therefore a(7) = #{7, 5+2, 5+1+1, 4+2+1, 4+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1} = 9.

R. Zumkeller, Table of n, a(n) for n = 1..100
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A160001, A000041, A159999, A006370.

Cf. A070165.

a160000 n = p (takeWhile (<= n) $ sort $ a070165_row n) n where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Sep 01 2012

A160001 Number of partitions of n into distinct parts occurring in '3x+1'-trajectory starting with n.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 3, 1, 4, 3, 4, 9, 4, 8, 5, 1, 6, 18, 10, 5, 2, 13, 6, 25, 19, 10, 8, 41, 24, 14, 10, 1, 40, 19, 8, 138, 90, 55, 62, 7, 17, 3, 144, 63, 34, 16, 16, 43, 218, 148, 105, 22, 7, 39, 24, 323, 371, 150, 224, 54, 27, 40, 30, 1, 572, 444, 344, 71, 32, 19, 39, 1766, 52

Offset: 1

Reinhard Zumkeller, May 11 2009

nonn

			7-22-11-34-17-52-26-13-40-20-10-5-16-8-[4-2-1]*:
{1,2,4,5,7} is the set of numbers <= n, occurring in this trajectory, therefore a(7) = #{7, 5+2, 4+2+1} = 3.

R. Zumkeller, Table of n, a(n) for n = 1..100
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A160000, A000009, A159999, A006370.

Cf. A070165.

a160001 n = p (takeWhile (<= n) $ sort $ a070165_row n) n where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Sep 01 2012

A214614 Irregular triangle read by rows: row n gives numbers <= n whose Collatz trajectory contains the trajectory of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 4, 5, 7, 1, 2, 4, 8, 1, 2, 4, 5, 7, 8, 9, 1, 2, 4, 5, 8, 10, 1, 2, 4, 5, 8, 10, 11, 1, 2, 3, 4, 5, 6, 8, 10, 12, 1, 2, 4, 5, 8, 10, 13, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 1, 2, 4, 5, 8, 10, 15

Offset: 1

Jayanta Basu, Mar 06 2013

Each row has A159999(n) elements and ends in n.

			Rows of triangle:
{1},
{1, 2},
{1, 2, 3},
{1, 2, 4},
{1, 2, 4, 5},
{1, 2, 3, 4, 5, 6},
{1, 2, 4, 5, 7},
{1, 2, 4, 8},
{1, 2, 4, 5, 7, 8, 9},
{1, 2, 4, 5, 8, 10}

Cf. A070165, A159999.

import Data.List (sort)
a214614 n k = a214614_tabf !! (n-1) (k-1)
a214614_row n = a214614_tabf !! (n-1)
a214614_tabf = zipWith f [1..] a070165_tabf where
                       f v ws = sort $ filter (<= v) ws
-- Reinhard Zumkeller, Sep 01 2014

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; f[n_] := Module[{c = Collatz[n]}, Select[c, # <= n &]]; t = Table[f[n], {n, 20}]; Flatten[t] (* T. D. Noe, Mar 07 2013 *)

A246436 Number of numbers 1,...,n not occurring in the Collatz trajectory starting with n.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 4, 2, 4, 4, 3, 6, 4, 8, 11, 8, 4, 8, 12, 15, 10, 14, 13, 11, 16, 17, 12, 15, 19, 21, 26, 16, 22, 26, 16, 19, 21, 21, 31, 28, 34, 23, 28, 31, 34, 34, 36, 29, 29, 33, 40, 43, 36, 40, 36, 33, 39, 37, 44, 47, 45, 48, 57, 42, 41, 45, 53, 56

Offset: 1

Reinhard Zumkeller, Sep 01 2014

nonn

a(n) = n - A159999(n).

			8-4-2-1: a(8) = #{3,5,6,7} = 4;
9-28-14-7-22-11-34-17-52-26-13-40-20-10-5-16-8-4-2-1: a(9) = #{3,6} = 2;
10-5-16-8-4-2-1: a(10) = #{3,6,7,9} = 4;
11-34-17-52-26-13-40-20-10-5-16-8-4-2-1: a(11) = #{3,6,7,9} = 4;
12-6-3-10-5-16-8-4-2-1: a(12) = #{7,9,11} = 3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A159999, A220237, A070165.

import Data.List ((\\), genericIndex)
a246436 n = length $ [1..n] \\ genericIndex a220237_tabf (n - 1)

Table[Length[Complement[Range[n],NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]],{n,70}] (* Harvey P. Dale, May 27 2018 *)

cp's OEIS Frontend

A008908 a(n) = (1 + number of halving and tripling steps to reach 1 in the Collatz (3x+1) problem), or -1 if 1 is never reached.

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

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A076228 Number of terms k in the trajectory of the Collatz function applied to n such that k < n.

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A160000 Number of partitions of n into parts occurring in '3x+1'-trajectory starting with n.

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A160001 Number of partitions of n into distinct parts occurring in '3x+1'-trajectory starting with n.

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Haskell

A214614 Irregular triangle read by rows: row n gives numbers <= n whose Collatz trajectory contains the trajectory of n.

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Mathematica

A246436 Number of numbers 1,...,n not occurring in the Collatz trajectory starting with n.

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