Let $g_n$ be the $n$ th prime gap.
Let $f_n = max( g_1,g_2,...,g_n)$
Now take the sequence $f_n$ and remove the duplicates. Also sort from small to large.
Then $f_n $ gives the sequence
This sequence is somewhat mysterious to me.
For instance I expected at first $ f_1 = 1, f_m = 2m - 2$ for all $m > 1$.
But here we see a “ jump “ from $8$ to $14$.
I know alot has been investigated surrounding prime gaps and many things are proved ( or disproved ). Yet also many things are not proved at all. The connections between the open problems is Also Sometimes unclear.
So I wonder about the sequence $f_n$ and How it relates to all that. Many related functions can be defined.
Let $t_n$ be the $n$ th even number not in $f_n$.
So the sequence starts $10,12,...$
How fast does $t_n$ grow ??
Lists and plots would be Nice too.
I thought the hardy-littlewood conjecture about the density of prime gaps of fixed gap size would solve my questions.
But I was unable to deduce much.
I Also tried to combine hardy-littlewood with cramers conjecture and the generalised riemann hypothesis but also without luck.
Since the average prime gap $g_n$ is asymptoticly to $ ln(n) $ and the geometric average is asymptotic to $ C \cdot ln(n) $ , these jumps surprise me. ( or did I miss something Then ?? )
I need more information on this.
Main question :
Is $t_n$ a finite sequence ??