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