Sequence of Successive Longest Collatz Sequence Generators

Theorem

The sequence of integers which generate a Collatz process which is longer than that of any smaller integers begins:

$\ds 1$

$:$

$\ds $

$0$ steps

$\ds 2$

$:$

$\ds $

$1$ step

$\ds 3$

$:$

$\ds $

$7$ steps

$\ds 6$

$:$

$\ds $

$8$ steps

$\ds 7$

$:$

$\ds $

$16$ steps

$\ds 9$

$:$

$\ds $

$19$ steps

$\ds 18$

$:$

$\ds $

$20$ steps

$\ds 25$

$:$

$\ds $

$23$ steps

$\ds 27$

$:$

$\ds $

$111$ steps

This sequence is A006877 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

Proof

Missing complementary orbits:

$\ds 4$

$:$

$\ds $

$2$ steps

$\ds 5$

$:$

$\ds $

$5$ steps

$\ds 8$

$:$

$\ds $

$3$ steps

$\ds 10$

$:$

$\ds $

$6$ steps

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Sources

1997: David Wells: Curious and Interesting Numbers (2nd ed.) ... (previous) ... (next): $27$