A192719 Chain of Collatz sequences.

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

1,2

Comments

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

Links

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

Example

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;
...

Prog

(Java) See Lyons link.
(Haskell)
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

Crossrefs

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

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

Sequence in context: A260758 A091858 A070165 * A270996 A366325 A203709

Adjacent sequences: A192716 A192717 A192718 * A192720 A192721 A192722

Keyword

nonn,easy,tabf

Author

Robert C. Lyons, Dec 31 2012

Status

approved