A222118 -id:A222118 - cp's OEIS frontend

A359215 Number of terms in S(n) that did not appear in previous trajectories, where S(n) is the trajectory of the mappings of x->A359194(x) starting with n and stopping when 0 is reached, -1 if 0 is never reached.

0, 1, 1, 11, 1, 1, 0, 2, 1, 1, 0, 6, 78, 0, 2, 0, 0, 1, 0, 1, 1, 0, 0, 3, 0, 0, 11, 0, 7571, 2, 0, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 1, 0, 1, 3, 0, 3, 77, 0, 5419, 1, 0, 1, 4, 0, 1, 0, 0, 2, 2, 0, 2, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1

Offset: 0

Michael De Vlieger, Dec 21 2022

"Branch length" of n->A359194(n).
a(0) = 0 since n = 0.
Let m be the first term in S(n) that has appeared in S(k), k < n. A359218(n) = m.
Analogous to A222118 which instead regards the Collatz function A006318.

			a(0) = 0 since n = 0.
a(1) = 1 since S(1) = {1, 0}, but m = 0 appeared in S(0).
a(2) = 1 since S(2) = {2, 1, 0}, but m = 1 appeared in S(1).
a(3) = 11 since S(3) = {3, 6, 13, 24, 55, 90, 241, 300, 123, 142, 85, 0}, but m = 0 appeared in S(0).
a(4) = 1 since S(4) = {4, 3, ...} but 3 appears in S(3), etc.

Michael De Vlieger, Table of n, a(n) for n = 0..16384
Michael De Vlieger, Log log scatterplot of a(n) and b(n), n = 1..2^14, b(n) = A359207(n) in dark blue, a(n) in red, highlighting where a(n) = b(n) in green.

Cf. A222118, A359194, A359207, A359218.

c[] = -1; c[0] = 0; f[n] := FromDigits[BitXor[1, IntegerDigits[3*n, 2]], 2]; Table[(Map[If[c[#1] == -1, Set[c[#1], #2]] & @@ # &, Partition[#, 2, 1]]; -1 + Length[#]) &@ NestWhileList[f, n, c[#] == -1 &], {n, 0, 120}]

A263716 Irregular triangle read by rows: numbers in the Collatz conjecture in the order of their first appearance.

Original entry on oeis.org

1, 2, 3, 10, 5, 16, 8, 4, 6, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 9, 28, 14, 12, 15, 46, 23, 70, 35, 106, 53, 160, 80, 18, 19, 58, 29, 88, 44, 21, 64, 32, 24, 25, 76, 38, 27, 82, 41, 124, 62, 31, 94, 47, 142, 71, 214, 107, 322, 161, 484, 242, 121, 364, 182

Offset: 0

Daniel Suteu, Oct 24 2015

This is the irregular triangle read by rows giving trajectory of n in the Collatz problem, flattened and with all the repeated terms deleted.

This sequence goes to infinity as n gets larger. On the Collatz conjecture this sequence is a permutation of the positive integers. [Corrected by Charles R Greathouse IV, Jul 29 2016]

			Triangle begins:
1;
2;
3, 10, 5, 16, 8, 4;
...
The Collatz trajectories for the first five positive integers are {1}, {2, 1}, {3, 10, 5, 16, 8, 4, 2, 1}, {4, 2, 1}, {5, 16, 8, 4, 2, 1}.
From {2, 1} we delete 1 because it has already occurred. From {3, 10, 5, ..., 4, 2, 1} we delete {2, 1} because both numbers have already occurred. We completely get rid of {4, 2, 1} because it has already occurred as the tail end of {3, 10, 5, ...}, and we also completely get rid of {5, 16, 8, ...} for the same reason.
This leaves us with {1}, {2}, {3, 10, 5, 16, 8, 4}, thus accounting for the first eight terms of this sequence.

Cf. A006577, A070165, A222118 (row lengths).

Cf. A347265 (essentially the same).

collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; DeleteDuplicates[Flatten[Table[collatz[n], {n, 20}]]] (* Alonso del Arte, Oct 24 2015 *)

func collatz(n) is cached {  # automatically memoized function
    say n;                   # prints the first unseen numbers
    n.is_one ? 0
             : (n.is_even ? collatz(n/2)
                          : collatz(3*n + 1));
}
range(1, Math.inf).each { |i| collatz(i) }

row(n) = {
   if seen[n]: stop
   else: write(n) and do:
   | n is one: stop
   | n is odd: n <- 3*n+1
   | n is even: n <- n/2
}

A221956 Number of values in the Collatz (3x+1) trajectory starting at n that are also present in the trajectory for some k < n.

Original entry on oeis.org

0, 1, 2, 3, 6, 8, 7, 4, 17, 7, 15, 9, 10, 18, 9, 5, 13, 20, 16, 8, 5, 16, 16, 10, 21, 11, 17, 19, 19, 18, 107, 6, 24, 14, 14, 21, 19, 22, 23, 9, 110, 8, 22, 17, 14, 17, 105, 11, 25, 25, 20, 12, 12, 112, 106, 20, 30, 20, 33, 19, 20, 108, 95, 7, 28, 27, 28, 15, 12

Offset: 1

Michel Lagneau, Feb 23 2013

			a(15) = 9 because the Collatz trajectory starting at 15 is (15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1), which has 9 values, namely, 40, 20, 10, 5, 16, 8, 4, 2, and 1, in common with the trajectory for at least one k < 15.

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

Cf. A006577, A222118.

lst1:={1}: L:={1}:for n from 2 to 200 do: x := n :lst:={n}:while x > 1 do if type(x, 'even') then x := x/2:lst:=lst union {x}:  else x := 3*x+1 : lst:=lst union {x}: end if; end do;lst2:=L intersect lst:n2:=nops(lst2 ): printf(`%d, `,n2): L:=L union lst:od:

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

A222297 Irregular triangle of numbers first appearing in the Collatz (3x+1) iteration of n.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 10, 16, 6, 7, 11, 13, 17, 20, 22, 26, 34, 40, 52, 9, 14, 28, 12, 15, 23, 35, 46, 53, 70, 80, 106, 160, 18, 19, 29, 44, 58, 88, 21, 32, 64, 24, 25, 38, 76, 27, 31, 41, 47, 61, 62, 71, 82, 91, 92, 94, 103, 107, 121, 122, 124, 137, 142, 155, 161

Offset: 1

T. D. Noe, Feb 23 2013

Sequence A222118 gives the number of terms in each row.

			The irregular table begins
{1},
{2},
{3, 4, 5, 8, 10, 16},
{},
{},
{6},
{7, 11, 13, 17, 20, 22, 26, 34, 40, 52},
{},
{9, 14, 28},
{},
{},
{12},
{},
{},
{15, 23, 35, 46, 53, 70, 80, 106, 160}

T. D. Noe, Rows n = 1..4000 of irregular triangle, flattened

Cf. A222118.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; found = {}; t = Table[c = Collatz[n]; r = Complement[c, found]; found = Union[found, c]; r, {n, 27}]; Flatten[t]

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 *)

cp's OEIS Frontend

A359215 Number of terms in S(n) that did not appear in previous trajectories, where S(n) is the trajectory of the mappings of x->A359194(x) starting with n and stopping when 0 is reached, -1 if 0 is never reached.

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A263716 Irregular triangle read by rows: numbers in the Collatz conjecture in the order of their first appearance.

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Sidef

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A221956 Number of values in the Collatz (3x+1) trajectory starting at n that are also present in the trajectory for some k < n.

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A222297 Irregular triangle of numbers first appearing in the Collatz (3x+1) iteration of n.

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

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