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Consider the $3n+1$ sequence.

Let be $\sigma(n)$ the Number of steps necessary to reach the maximum of the trajectory starting from an integer $n$.

Let $\tau(n)$ be the Number of steps necessary to reach $1$ starting from $n$, I think that this is called the delay.

Now consider $\sigma(n)-\tau(n)=\gamma(n)$, that is the number of steps from the maximum to $1$.

Now consider $\xi(n)=\frac{\gamma(n)}{n}$

Now I call an integer $N$ a $\xi$ -record if for all $n<N$ we have $\xi(N)<\xi(n)$

a) Does this produce a sequence of records?
b) Can this sequence be useful for the $3n+1$ problem?

c) Is there already an OEIS sequence?

Gottfried Helms
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    You would have to work out some initial terms to check in OEIS. Another "Collatz records" sequence there is OEIS A025587, based on blowout factors for different starting points - the highest ratio of maximum term reached to starting point. – Henry Oct 22 '23 at 12:55
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    Do you have some data examples? Perhaps provide some? – Gottfried Helms Oct 22 '23 at 18:00
  • you'll probably find powers of $2$ in that sequence, since it maximize $N$ and minimize $\gamma(N)$ – Collag3n Oct 22 '23 at 20:27
  • Hmm, isn't $\tau(n)$ always larger than $\sigma(n)$, so $\gamma(n)$ is always negative? – Gottfried Helms Oct 23 '23 at 09:40
  • @Gottfried Helms, since it was described as "the number of steps", and I see no reason to make it a negative number, I assumed it to be positive (and that the inversion was just a small mistake). But now that you ask, I am wondering if it was intentional.... – Collag3n Oct 23 '23 at 11:23
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    and the record $7$ with $|\gamma(N)|=1.571428...$ seems hard to beat (any link with $\log_2(3)=1.5849...$?) – Collag3n Oct 23 '23 at 11:42

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