A177729 -id:A177729 - cp's OEIS frontend

A220263 Lengths of Collatz trajectories starting with terms of A177729.

1, 2, 8, 9, 17, 20, 10, 18, 21, 21, 8, 11, 24, 112, 19, 27, 22, 22, 35, 9, 30, 17, 12, 25, 113, 113, 33, 20, 108, 28, 15, 23, 116, 15, 36, 36, 23, 10, 31, 18, 18, 13, 119, 26, 26, 39, 114, 114, 70, 34, 34, 21, 21, 47, 109, 47, 122, 29, 29, 42, 16, 16, 24

Offset: 1

Reinhard Zumkeller, Jan 03 2013

nonn

Row lengths in A192719.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A006577.

Haskell
```
a220263 = length . a192719_row
```

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; t={1}; Do[If[FreeQ[Union@@Table[coll[i],{i,n-1}],n],AppendTo[t,Length[coll[n]]]],{n,2,144}]; t (* Jayanta Basu, May 29 2013 *)

A061641 Pure numbers in the Collatz (3x+1) iteration. Also called pure hailstone numbers.

Original entry on oeis.org

0, 1, 3, 6, 7, 9, 12, 15, 18, 19, 21, 24, 25, 27, 30, 33, 36, 37, 39, 42, 43, 45, 48, 51, 54, 55, 57, 60, 63, 66, 69, 72, 73, 75, 78, 79, 81, 84, 87, 90, 93, 96, 97, 99, 102, 105, 108, 109, 111, 114, 115, 117, 120, 123, 126, 127, 129, 132, 133, 135, 138, 141, 144, 145

Offset: 1

Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 14 2001

Let {f(k,N), k=0,1,2,...} denote the (3x+1)-sequence with starting value N; a(n) denotes the smallest positive integer which is not contained in the union of f(k,0),...,f(k,a(n-1)).
In other words, a(n) is the starting value of the next '3x+1'-sequences in the sense that a(n) is not a value in any sequence f(k,N) with N < a(n).
f(0,N)=N, f(k+1,N)=f(k,N)/2 if f(k,N) is even and f(k+1,N)=3*f(k,N)+1 if f(k,N) is odd.
For all n, a(n) mod 6 is 0, 1 or 3. I conjecture that a(n)/n -> C=constant for n->oo, where C=2.311...
The Collatz conjecture says that for all positive n, there exists k such that C_k(n) = 1. Shaw states [p. 195] that "A positive integer n is pure if its entire tree of preimages under the Collatz function C are greater than or equal to it; otherwise n is impure. Equivalently, a positive integer n is impure if there exists rGary W. Adamson, Jan 28 2007
Pure numbers remaining after deleting the impure numbers in the hailstone (Collatz) problem; where the operation C(n) = {3n+1, n odd; n/2, n even}. Add the 0 mod 3 terms in order, among the terms of A127633, since all 0 mod 3 numbers are pure. - Gary W. Adamson, Jan 28 2007
After computing all a(n) < 10^9, the ratio a(n)/n appears to be converging to 2.31303... Hence it appears that the numbers in this sequence have a density of about 1/3 (due to all multiples of 3) + 99/1000. - T. D. Noe, Oct 12 2007
A016945 is a subsequence. - Reinhard Zumkeller, Apr 17 2008

			Consider n=3: C(n), C_2(n), C_3(n), ...; the iterates are 10, 5, 16, 8, 4, 2, 1, 4, 2, 1; where 4, 5, 8, 10 and 16 have appeared in the orbit of 3 and are thus impure.
a(1)=1 since Im(f(k,0))={0} for all k and so 1 is not a value of f(k,0). a(2)=3 since Im(f(k,0)) union Im(f(k,1))={0,1,2,4} and 3 is the smallest positive integer not contained in this set.

T. D. Noe, Table of n, a(n) for n = 1..10000
Douglas J. Shaw, The Pure Numbers Generated by the Collatz Sequence, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, pp. 194-201.
B. Snapp and M. Tracy, The Collatz Problem and analogues, JIS 11 (2008) 08.4.7.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A070165 (Collatz trajectories), A127633, A336938, A336938. See A177729 for a variant.

DoCollatz[n_] := Module[{m = n}, While[m > nn || ! reached[[m]], If[m <= nn, reached[[m]] = True]; If[EvenQ[m], m = m/2, m = 3 m + 1]]]; nn = 200; reached = Table[False, {nn}]; t = {0, 1}; While[DoCollatz[t[[-1]]]; pos = Position[reached, False, 1, 1]; pos != {}, AppendTo[t, pos[[1, 1]]]]; t (* T. D. Noe, Jan 22 2013 *)

firstMiss(A) = { my(i); if(#A == 0 || A[1] > 0, return(0)); for(i = 1, A[#A] + 1, if(!setsearch(A,i), return(i))); };
iter(A) = { my(a = firstMiss(A)); while(!setsearch(A,a), A = setunion(A, Set([a])); a = if(a % 2, 3*a+1, a/2)); A; };
makeVec(m) = { my(v = [], A = Set([]), i); for(i = 1, m, v = concat(v, firstMiss(A)); if (i < m, A = iter(A))); v; };
makeVec(64) \\ Markus Sigg, Aug 08 2020

Edited by T. D. Noe and N. J. A. Sloane, Oct 16 2007

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

A187831 Smallest number m > n such that n occurs in Collatz trajectory starting with m; a(0) = 1 by convention.

Original entry on oeis.org

1, 2, 3, 6, 5, 6, 12, 9, 9, 18, 11, 14, 24, 14, 18, 30, 17, 18, 36, 25, 22, 42, 25, 27, 48, 33, 28, 54, 36, 33, 60, 41, 42, 66, 36, 41, 72, 43, 39, 78, 41, 54, 84, 57, 50, 90, 47, 54, 96, 57, 66, 102, 56, 54, 108, 73, 57, 114, 59, 78, 120, 62, 82, 126, 75

Offset: 0

Reinhard Zumkeller, Jan 04 2013

nonn

A070165(a(n),k) = n for some k with 1 <= k <= A006577(a(n)).

			n = 10: row 11 of A070165 = [11,34,17,52,26,13,40,20,10,5,16,8,4,2,1],
therefore A070165(11,9) = 10 and a(10) = 11;
n = 11: rows 12 and 13 of A070165 don't contain 11, but 14 does:
row 12: [12,6,3,10,5,16,8,4,2,1],
row 13: [13,40,20,10,5,16,8,4,2,1],
row 14: [14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1],
therefore A070165(14,4) = 11: a(11) = 14.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A177729, A061641.

Cf. A225840.

import Data.List (find)
import Data.Maybe (fromJust)
a187831 0 = 1
a187831 n = head $ fromJust $
        find (n `elem`) $ drop (fromIntegral n) a070165_tabf

mcollQ[n_,k_]:=MemberQ[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],k]==True; Prepend[Table[i=n+1; While[!mcollQ[i,n],i++]; i,{n,64}],1] (* Jayanta Basu, May 27 2013 *)
Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Join[{1}, Table[k = n + 1; While[! MemberQ[Collatz[k], n], k++]; k, {n, 100}]] (* T. D. Noe, May 28 2013 *)

A359226 a(n) is the least k >= 0 such that A006370^k(A070167(n)) = n (where A006370^k denotes the k-th iterate of A006370).

Original entry on oeis.org

0, 0, 0, 5, 2, 0, 0, 4, 0, 1, 2, 0, 7, 2, 0, 3, 4, 0, 0, 9, 0, 1, 2, 0, 0, 6, 0, 1, 2, 0, 5, 2, 0, 3, 4, 0, 0, 2, 0, 8, 2, 0, 0, 4, 0, 1, 7, 0, 5, 2, 0, 5, 6, 0, 0, 2, 0, 1, 2, 0, 92, 4, 0, 1, 2, 0, 7, 2, 0, 3, 9, 0, 0, 7, 0, 1, 2, 0, 0, 8, 0, 1, 2, 0, 5, 2, 0

Offset: 1

Rémy Sigrist, Dec 22 2022

nonn

If we travel the Collatz tree backwards, we will observe no further branching, whenever we have reached a number divisible by three. This is the reason why a(n) will be zero in all cases where n is divisible by three, because in this case no smaller number can exist further upward in the Collatz tree, as the actual branch will progress in numbers of the form 3*m*2^k. - Thomas Scheuerle, Dec 22 2022

			The Collatz sequence starting from 1 is: 1.
So a(1) = 0.
The Collatz sequence starting from 2 is: 2, 1.
So a(2) = 0.
The Collatz sequence starting from 3 is: 3, 10, 5, 16, 8, 4, 2, 1.
So a(3) = 0, a(10) = 1, a(5) = 2, a(16) = 3, a(8) = 4, a(4) = 5.

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, PARI program
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070167, A177729.

nn = 87; c[] = -1; c[0] = 0; Do[(MapIndexed[If[c[#1] == -1, Set[c[#1], First[#2] - 1]] &, #]; -1 + Length[#]) &@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, c[#] == -1 && # > 1 &], {n, 0, nn}]; Array[c, nn] (* _Michael De Vlieger, Dec 23 2022 *)

PARI
```
See Links section.
```

a(n) = 0 iff n belongs to A177729.

If a(n) > 0 and n is odd, then a(n*3+1) - a(n) = 1. If a(n) > 0 and n is even, then a(n*3+1) - a(n*6+2) = 1. - Thomas Scheuerle, Dec 22 2022

A213672 Final term in Collatz trajectory of n that did not appear in previous trajectories.

Original entry on oeis.org

1, 2, 4, 0, 0, 6, 20, 0, 14, 0, 0, 12, 0, 0, 80, 0, 0, 18, 44, 0, 32, 0, 0, 24, 38, 0, 92, 0, 0, 30, 0, 0, 50, 0, 0, 36, 56, 0, 152, 0, 0, 42, 74, 0, 68, 0, 0, 48, 0, 0, 116, 0, 0, 54, 188, 0, 86, 0, 0, 60, 0, 0, 728, 0, 0, 66, 0, 0, 104, 0, 0, 72, 110, 0, 128, 0

Offset: 1

Jayanta Basu, Mar 03 2013

nonn

This can be considered as the step down value, as beyond this point the trajectory of n reduces to a lower trajectory. When n is impure we define a(n)=0; see also A177729.

			a(7)=20 because after 20 trajectory of 7 becomes identical with trajectory of 3.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A177729, A222118.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; prev = {}; Table[c = Collatz[n]; If[Complement[c, prev] == {}, 0, i = 1; While[MemberQ[prev, c[[-i]]], i++]; prev = Union[prev, c]; c[[-i]]], {n, 100}] (* T. D. Noe, Mar 03 2013 *)

A235452 Take the union of all the sequences Collatz(i) for i <= n. The number a(n) is the largest of consecutive numbers beginning with 1.

Original entry on oeis.org

1, 2, 5, 5, 5, 6, 8, 8, 11, 11, 11, 14, 14, 14, 17, 17, 17, 18, 20, 20, 23, 23, 23, 24, 26, 26, 29, 29, 29, 32, 32, 32, 35, 35, 35, 36, 38, 38, 41, 41, 41, 42, 44, 44, 47, 47, 47, 50, 50, 50, 53, 53, 53, 54, 56, 56, 59, 59, 59, 62, 62, 62, 65, 65, 65

Offset: 1

Martin Y. Champel, Jan 10 2014

nonn

The Collatz sequence is also called the 3x+1 sequence.

			Let the C(n) function compute the Collatz sequence starting at n.
For n = 1, C(1) = {1} then term 1 is 1.
For n = 2, C(2) = {1,2} then term 2 is 2.
For n = 3, C(3) = {3,10,5,16,8,4,2,1} = {1,2,3,4,5,8,10,16} then it contains {1,2,3,4,5} but not {1,2,3,4,5,6} then term 3 is 5.
For n = 4, C(4) = C(3) then term 4 is 5.
For n = 5, C(5) = C(4) = C(3) then term 5 is 5.
For n = 6, C(6) = {1,2,3,4,5,6,8,10,16} then term 6 is 6.

Martin Y. Champel, Table of n, a(n) for n = 1..1000

Cf. A061641, A177729.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countConsec[lst_] := Module[{cnt = 0, i = 1}, While[i <= Length[lst] && lst[[i]] == i, cnt++; i++]; cnt]; mx = 0; u = {}; Table[c = Collatz[n]; u = Union[u, c]; mx = Max[mx, countConsec[u]], {n, 65}] (* T. D. Noe, Feb 23 2014 *)

def A235452(n=100):
    a = set([])
    A235452 = {1: 1}
    for i in range(2, n):
        c = i
        a.add(c)
        while c != 1:
            if c % 2 == 1:
                c = 3 * c + 1
                a.add(c)
            c = c / 2
            a.add(c)
        k = 1
        while k in a:
            k += 1
        A235452[i] = k - 1
    return A235452
seq_map = A235452()
for n in range(1, len(seq_map) + 1):
    print(seq_map[n], end=", ")

A318613 Length of iterations of positive integers which do not appear in a Collatz sequence starting from a smaller positive integer.

Original entry on oeis.org

0, 1, 7, 8, 16, 19, 9, 17, 20, 20, 7, 10, 23, 111, 18, 26, 21, 21, 34, 8, 29, 16, 11, 24, 112, 112, 32, 19, 107, 27, 14, 22, 115, 14, 35, 35, 22, 9, 30, 17, 17, 12, 118, 25, 25, 38, 113, 113, 69, 33, 33, 20, 20, 46, 108, 46, 121, 28, 28, 41, 15, 15

Offset: 1

Peter Stikker, Aug 30 2018

nonn

The sequence of the length of the iteration in the Collatz sequence is known as A006577, and the sequence of positive integers which do not appear in a Collatz sequence starting from a smaller positive integer is listed as A177729. This sequence is basically the combination of the two. Taking the sequence A177729 as input and then listing the length of the iteration for those (from A006577).

Cf. A177729, A006577.

Almost the same as A127933.

cp's OEIS Frontend

A220263 Lengths of Collatz trajectories starting with terms of A177729.

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A061641 Pure numbers in the Collatz (3x+1) iteration. Also called pure hailstone numbers.

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Extensions

A192719 Chain of Collatz sequences.

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

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A187831 Smallest number m > n such that n occurs in Collatz trajectory starting with m; a(0) = 1 by convention.

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A359226 a(n) is the least k >= 0 such that A006370^k(A070167(n)) = n (where A006370^k denotes the k-th iterate of A006370).

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A213672 Final term in Collatz trajectory of n that did not appear in previous trajectories.

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Mathematica

A235452 Take the union of all the sequences Collatz(i) for i <= n. The number a(n) is the largest of consecutive numbers beginning with 1.