A350369 a(n) is the length of the longest sequence of consecutive tripling steps in the Collatz (3x+1) sequence beginning at n.

0, 0, 2, 0, 1, 2, 3, 0, 3, 1, 2, 2, 1, 3, 4, 0, 1, 3, 2, 1, 1, 2, 3, 2, 2, 1, 6, 3, 2, 4, 6, 0, 2, 1, 2, 3, 3, 2, 3, 1, 6, 1, 3, 2, 1, 3, 6, 2, 3, 2, 2, 1, 1, 6, 6, 3, 3, 2, 2, 4, 3, 6, 6, 0, 3, 2, 2, 1, 1, 2, 6, 3, 6, 3, 2, 2, 2, 3, 4, 1, 3, 6, 6, 1, 1, 3, 3

Offset

1,3

Comments

"Consecutive tripling steps" are repeated (3x+1)/2 operations that are not interrupted by a second division by 2.

This sequence attempts to measure the largest upward thrust in each Collatz sequence and so is correlated to some degree with the maximum value (A025586) and length (A006577) of Collatz sequences.

If n = 2^x * (2^y*z - 1), then a(n) >= y. - Charles R Greathouse IV, Oct 25 2022

Links

Kevin P. Thompson, Table of n, a(n) for n = 1..10000

Index entries for sequences related to 3x+1 (or Collatz) problem

Example

The Collatz sequence for n=7 has a streak of 3 consecutive tripling steps (at 7, 11, and 17), so a(7) = 3.
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
^      ^       ^

Prog

(PARI) a(n)=my(c, r); n>>=valuation(n, 2); while(n>1, n+=(n+1)/2; if(n%2, c++, r=max(r, c+1); n>>=valuation(n, 2); c=0)); max(r, c) \\ Charles R Greathouse IV, Oct 25 2022

Crossrefs

Cf. A006370, A006577, A006667, A025586, A213215, A221469, A347270, A347409.

Sequence in context: A263833 A308625 A221469 * A117398 A295989 A240852

Adjacent sequences: A350366 A350367 A350368 * A350370 A350371 A350372

Keyword

nonn,easy

Author

Kevin P. Thompson, Dec 27 2021

Status

approved