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?