A070165 -id:A070165 - cp's OEIS frontend

A350279 Irregular triangle T(n,k) read by rows in which row n lists the iterates of the Farkas map (A349407) from 2*n - 1 to 1.

1, 3, 1, 5, 3, 1, 7, 11, 17, 9, 3, 1, 9, 3, 1, 11, 17, 9, 3, 1, 13, 7, 11, 17, 9, 3, 1, 15, 5, 3, 1, 17, 9, 3, 1, 19, 29, 15, 5, 3, 1, 21, 7, 11, 17, 9, 3, 1, 23, 35, 53, 27, 9, 3, 1, 25, 13, 7, 11, 17, 9, 3, 1, 27, 9, 3, 1, 29, 15, 5, 3, 1

Offset: 1

Paolo Xausa, Dec 22 2021

			Written as an irregular triangle, the sequence begins:
  n\k   1   2   3   4   5   6   7
  -------------------------------
   1:   1
   2:   3   1
   3:   5   3   1
   4:   7  11  17   9   3   1
   5:   9   3   1
   6:  11  17   9   3   1
   7:  13   7  11  17   9   3   1
   8:  15   5   3   1
   9:  17   9   3   1
  10:  19  29  15   5   3   1
  11:  21   7  11  17   9   3   1
  12:  23  35  53  27   9   3   1

Paolo Xausa, Table of n, a(n) for n = 1..12301 (rows 1..1000 of triangle, flattened).
H. M. Farkas, "Variants of the 3N+1 Conjecture and Multiplicative Semigroups", in Entov, Pinchover and Sageev, Geometry, Spectral Theory, Groups, and Dynamics, Contemporary Mathematics, vol. 387, American Mathematical Society, 2005, p. 121.
J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, p. 74.

Cf. A349407, A375909 (# of iterations), A375910 (row sums), A375911 (row maxs).

Cf. A070165.

FarkasStep[x_] := Which[Divisible[x, 3], x/3, Mod[x, 4] == 3, (3*x + 1)/2, True, (x + 1)/2];
Array[Most[FixedPointList[FarkasStep, 2*# - 1]] &, 15] (* Paolo Xausa, Sep 03 2024 *)

T(n,1) = 2*n-1; T(n,k) = A349407((T(n,k-1)+1)/2), where n >= 1 and k >= 2.

A070991 Numbers n such that the trajectory of n under the `3x+1' map reaches n - 1.

Original entry on oeis.org

2, 3, 5, 6, 9, 11, 14, 17, 18, 39, 41, 47, 54, 57, 59, 62, 71, 81, 89, 107, 108, 161, 252, 284, 378, 639, 651, 959, 977, 1368, 1439, 1823, 2159, 2430, 2735, 3239, 4103, 4617, 4859, 6155, 7289, 9233

Offset: 1

Benoit Cloitre and Boris Gourevitch (boris(AT)pi314.net), May 18 2002

From Collatz conjecture, the trajectory of n never reaches n again. Is this sequence finite?

There are no more terms < 10^9. - Donovan Johnson, Sep 22 2013

			Trajectory of 39 is: (118, 59, 178, 89, 268, 134, 67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1) and 39-1 = 38 is reached, hence 39 is in the sequence.

Cf. A070165 (Collatz trajectories), A219696, A221213, A070993.

a070991 n = a070991_list !! (n-1)
a070991_list = filter (\x -> (x - 1) `elem` a070165_row x) [1..]
-- Reinhard Zumkeller, Feb 22 2013

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Select[Range[100000], MemberQ[Collatz[#], # - 1] &] (* T. D. Noe, Feb 21 2013 *)

for(n=1,10000,s=n; t=0; while(s!=1,t++; if(s%2==0,s=s/2,s=3*s+1); if(s==n-1,print1(n,","); ); ))

A070993 Numbers n such that the trajectory of n under the "3x+1" map reaches n+1.

Original entry on oeis.org

3, 7, 9, 15, 19, 25, 33, 39, 51, 91, 121, 159, 166, 183, 243, 250, 333, 376, 411, 432, 487, 501, 649, 667, 865, 889, 975, 1153, 1185, 1299, 1335, 1731, 1779, 2307, 3643, 4857, 7287

Offset: 1

Benoit Cloitre and Boris Gourevitch (boris(AT)pi314.net), May 18 2002

nonn

From Collatz conjecture, the trajectory of n never reaches n again. Is this sequence finite? (it seems there are no further terms below 10^6).

There are no more terms < 10^9. - Donovan Johnson, Sep 22 2013

			Trajectory of 39 is (118, 59, 178, 89, 268, 134, 67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1) which contains 39+1=40, so 39 is in the sequence.

Cf. A070165 (Collatz trajectories), A221213, A222293, A070991.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Select[Range[100000], MemberQ[Collatz[#], # + 1] &] (* T. D. Noe, Feb 22 2013 *)

for(n=1,10000,s=n; t=0; while(s!=1,t++; if(s%2==0,s=s/2,s=3*s+1); if(s==n-1,print1(n,","); ); ))

Corrected by T. D. Noe, Oct 25 2006

A087226 LCM of terms in Collatz (3x+1) function initiated at n.

Original entry on oeis.org

1, 2, 240, 4, 80, 240, 1361360, 8, 12252240, 80, 194480, 240, 1040, 1361360, 4095840, 16, 17680, 12252240, 107158480, 80, 1344, 194480, 1365280, 240, 535792400, 1040, 44841486948146266934850832405421294927083491752830032389039800908293040266400

Offset: 1

Labos Elemer, Aug 28 2003

nonn

			n=9: list={9,28,14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1};
LCM = 2*2*2*2*3*3*5*7*11*13*17 = 12252240.

Cf. A006370.

Cf. A070165, A225784.

a087226 = foldl1 lcm . a070165_row  -- Reinhard Zumkeller, May 16 2013

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] Table[Apply[LCM, fpl[w]], {w, 1, 32}]
Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[LCM @@ Collatz[n], {n, 27}] (* T. D. Noe, May 15 2013 *)

A192719 Chain of Collatz sequences.

Original entry on oeis.org

1, 2, 1, 3, 10, 5, 16, 8, 4, 2, 1, 6, 3, 10, 5, 16, 8, 4, 2, 1, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 12, 6, 3, 10, 5, 16, 8, 4, 2, 1

Offset: 1

Robert C. Lyons, Dec 31 2012

The sequence is a chain of Collatz sequences. The first Collatz sequence in the chain is (1). Each of the subsequent Collatz sequences in the chain starts with the minimum positive integer that does not appear in the previous Collatz sequences. If the Collatz conjecture is true, then each Collatz sequence in the chain will end with 1, and the chain will include an infinite number of distinct Collatz sequences. If the Collatz conjecture is false, then the chain will end with the first Collatz sequence that does not converge to 1.

T(n,1) = A177729(n). - Reinhard Zumkeller, Jan 03 2013

			The first Collatz sequence in the chain is (1). The second Collatz sequence in the chain is (2, 1), which starts with 2, since 2 is the smallest positive integer that doesn't appear the first Collatz sequence. The third Collatz sequence in the chain is (3, 10, 5, 16, 8, 4, 2, 1), which starts with 3, since 3 is the smallest positive integer that doesn't appear the previous Collatz sequences.
Thus this irregular array starts:
1;
2,  1;
3, 10,  5, 16,  8,  4,  2,  1;
6,  3, 10,  5, 16,  8,  4,  2,  1;
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10,  5, 16,  8, 4,  2, 1;
9, 28, 14,  7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1;
...

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Robert C. Lyons, Chain of Collatz sequences generator program in Java
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A070167, A070165, A061641, A177729, A192719.

Cf. A220263 (row lengths); A070165, A220237.

a192719 n k = a192719_tabf !! (n-1) !! (k-1)
a192719_row n = a192719_tabf !! (n-1)
a192719_tabf = f [1..] where
   f (x:xs) = (a070165_row x) : f (del xs $ a220237_row x)
   del us [] = us
   del us'@(u:us) vs'@(v:vs) | u > v     = del us' vs
                             | u < v     = u : del us vs'
                             | otherwise = del us vs
-- Reinhard Zumkeller, Jan 03 2013

Java
```
See Lyons link.
```

A222118 Number of terms in Collatz (3x+1) trajectory of n that did not appear in previous trajectories.

Original entry on oeis.org

1, 1, 6, 0, 0, 1, 10, 0, 3, 0, 0, 1, 0, 0, 9, 0, 0, 1, 5, 0, 3, 0, 0, 1, 3, 0, 95, 0, 0, 1, 0, 0, 3, 0, 0, 1, 3, 0, 12, 0, 0, 1, 8, 0, 3, 0, 0, 1, 0, 0, 5, 0, 0, 1, 7, 0, 3, 0, 0, 1, 0, 0, 13, 0, 0, 1, 0, 0, 3, 0, 0, 1, 3, 0, 8, 0, 0, 1, 9, 0, 1, 0, 0, 1, 0, 0, 7

Offset: 1

Jayanta Basu, Feb 23 2013

nonn

For n > 2, n such that a(n) = 0 are termed impure (A134191), while n such that a(n) > 0 are termed pure (A061641). - T. D. Noe, Feb 23 2013
From Robert G. Wilson v, Feb 25 2017: (Start)
For a(n) to be equal to 0, n != 0 (mod 3),
For a(n) to be an even positive number, n = {3, 7} (mod 12),
For a(n) to be equal to 1, n = {0, 1, 2, 3, 6, 7, 9} (mod 12),
For a(n) to be equal to 3, n = {1, 3, 9} (mod 12),
For a(n) to be an odd number > 3, n = {3, 7} (mod 12).
[Note that the above conditions are necessary but not sufficient. - Editors, Dec 15 2017]
(End)
a(n) gives the number of new terms in the n-th row of A070165 (see A263716). - Andrey Zabolotskiy, Feb 27 2017

			a(7) = 10, since trajectory of 7 includes 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, which did not appear in earlier trajectories.

Cf. A006577, A061641, A070165, A134191, A177729, A221956, A222297, A263716.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; found = {}; Table[c = Collatz[n]; r = Complement[c, found]; found = Union[found, c]; Length[r], {n, 100}] (* T. D. Noe, Feb 23 2013 *)

s = set([1])
print(1)
for n in range(2, 100):
    m, r = n, 0
    while m not in s:
        s.add(m)
        m = (m//2 if m%2==0 else 3*m+1)
        r += 1
    print(r)
# Andrey Zabolotskiy, Feb 21 2017

a(n) = A006577(n) - A221956(n) + 1. - Michel Lagneau, Feb 23 2013

A232711 Conjectured list of numbers whose trajectory under the '5x+1' map eventually reaches 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 16, 19, 24, 30, 32, 38, 48, 51, 60, 64, 65, 76, 96, 97, 102, 120, 128, 130, 137, 152, 155, 163, 175, 192, 194, 204, 219, 240, 243, 256, 260, 274, 304, 307, 310, 326, 343, 350, 384, 388, 397, 408, 417, 429, 438, 480, 486, 491, 512

Offset: 1

Jon Perry, Nov 28 2013

nonn

This is conjectural in that there is no known proof that 7, 9, 11, etc. (see A267970) do not eventually cycle. - N. J. A. Sloane, Jan 23 2016
It appears that most numbers diverge, but nothing is known for certain.
Note that the computer programs do not actually calculate a complete list of "numbers k such that the Collatz-like map T: if x odd, x -> 5*x+1 and if x even, x -> x/2, when started at k, eventually reaches 1".

			Beginning with 15 we get the trajectory 15, 76, 38, 19, 96, 48, 24, 12, 6, 3, 16, 8, 4, 2, 1, so 15 is a term.

Dmitry Kamenetsky, Table of n, a(n) for n = 1..1000
Tomás Oliveira e Silva, The px+1 problem.
Index entries for sequences related to 3x+1 (or Collatz) problem

See A267969, A267970 for other trajectories under this map T.
Cf. A057688, A133419.
Cf. A070165 (usual Collatz iteration).

for (i=1;i<2000;i++) {
c=0;
n=i;
while (n>1) {c++;if (n%2==0) n/=2; else n=5*n+1;if (c>100) break;}
if (c<101) document.write(i+", ");
} (Warning: bad program - will not find all the terms. - N. J. A. Sloane, Jan 23 2016)

cli=Compile[{{n,Integer}},If[OddQ[n],5n+1,n/2]//Round,RuntimeAttributes->{Listable},Parallelization->True]; okQ[n]:=Length[NestWhileList[cli,n, #>1&, 1,200]]<200; Select[Range[550],okQ] (* Harvey P. Dale, May 28 2014 *) (Warning: bad program - will not find all the terms. - N. J. A. Sloane, Jan 23 2016)

Entry revised (corrected definition, added warnings to programs, deleted b-file) by N. J. A. Sloane, Jan 23 2016

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 *)

A216022 Largest number m such that the Collatz trajectory starting at n contains all numbers not greater than m.

Original entry on oeis.org

1, 2, 5, 2, 2, 6, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Reinhard Zumkeller, Sep 01 2012

nonn

a(n) <= 6; a(A007283(n)) = 6;
a(n) > 1 for n > 1; a(n) <> 3; a(n) <> 4; a(n) <> 5 for n > 3;
a(n) = A216059(n) - 1.
In the first 100000 terms, there are only 16 terms greater than 2, all of which but one are equal to 6. - Harvey P. Dale, Nov 29 2019

			n = 3->10->5->16->8->4->2->1 => {1_2_3_4_5 8 10 16}, a(3) = 5;
n = 4->2->1 => {1_2 4}, a(4) = 2;
n = 5->16->8->4->2->1 => {1_2 4 5 8 16}, a(5) = 2;
n = 6->3->10->5->16->8->4->2->1 => {1_2_3_4_5_6 8 10 16}, a(6) = 6.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070165.

import Data.List (sort)
a216022 = length .
   takeWhile (== 0) . zipWith (-) [1..] . sort . a070165_row

scoll[n_]:=Sort[NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &]]; Join[{1,2},Table[i=1; While[scoll[n][[i]]==i,i++]; i-1,{n,3,86}]] (* Jayanta Basu, May 27 2013 *)
Join[{1,2},Flatten[Table[Position[Differences[Sort[ NestWhileList[ If[ EvenQ[#],#/2,3#+1]&,n,#>1&]]], ?(#>1&),1,1],{n,90}]]] (* _Harvey P. Dale, Nov 29 2019 *)

A216059 Smallest number not in Collatz trajectory starting at n.

Original entry on oeis.org

2, 3, 6, 3, 3, 7, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 7, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3

Offset: 1

Reinhard Zumkeller, Sep 01 2012

nonn

a(n) <= 7; a(A007283(n)) = 7;
a(n) <> 4; a(n) <> 4; a(n) <> 6 for n > 3;
a(n) = A216022(n) + 1.

Reinhard Zumkeller, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070165.

import Data.List ((\\))
a216059  n = head $ enumFromTo 1 (maximum ts + 1) \\ ts
   where ts = a070165_row n

scoll[n_]:=Sort[NestWhileList[If[EvenQ[#],#/2,3#+1] &,n,#>1 &]]; Join[{2,3},Table[i=1; While[scoll[n][[i]]==i,i++]; i,{n,3,86}]] (* Jayanta Basu, May 27 2013 *)

cp's OEIS Frontend

A350279 Irregular triangle T(n,k) read by rows in which row n lists the iterates of the Farkas map (A349407) from 2*n - 1 to 1.

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A070991 Numbers n such that the trajectory of n under the `3x+1' map reaches n - 1.

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A070993 Numbers n such that the trajectory of n under the "3x+1" map reaches n+1.

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Mathematica

PARI

Extensions

A087226 LCM of terms in Collatz (3x+1) function initiated at n.

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Haskell

Mathematica

A192719 Chain of Collatz sequences.

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Java

A222118 Number of terms in Collatz (3x+1) trajectory of n that did not appear in previous trajectories.

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A232711 Conjectured list of numbers whose trajectory under the '5x+1' map eventually reaches 1.

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JavaScript

Mathematica

Extensions

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

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