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

Author

Robert C. Lyons, Dec 31 2012

Keywords

• nonn
• easy
• tabf

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

Examples

			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;
...
		

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

Crossrefs

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

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

Programs

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

Java

See Lyons link.