Sequence of Successive Longest Collatz Sequence Generators

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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



Sources