Question about prime gap records

Question

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

$$ 1,2,4,6,8,14,...$$

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 ??

Answer 1

The driving factor in these gaps are the gaps between coprimes to the Nth primorial $P_N\#$ (the product of the first N primes). It may be useful to read about Jacobsthal's Function, see here: jacobsthal

These gaps are symmetric about the primorials and have neat fractal properties. There is much literature on the upper bounds of Jacobsthal's function. Perhaps you may be able to find something in this literature to help whittle down your approach to $t_n$.

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With respect to fractals, define $C_N(k)$ to be the kth positive coprime to Nth primorial ($P_N\#$). Then, there exists an $M<N$ such that

$$ C_N(k+1)-C_N(k) = C_M(i)+C_M(j) $$

In connection to the prime gaps $g_n$, we have for $1<k<\pi(P_{N+1}^2)-N$ that

$$ C_N(k+1)-C_N(k)=g_{N+k-1} $$

Answer 2

Is tn a finite sequence ?? See https://en.wikipedia.org/wiki/Polignac%27s_conjecture list: http://www.trnicely.net/gaps/gaplist.html#MainTable

Answer 3

Let's retain the $f_n$ notation for the sequence of maximal gaps after removing the duplicates, then $$ \{f_n\}=\{1, 2, 4, 6, 8, 14, 18, 20, 22, 34, 36, 44, 52, 72, 86, 96, 112,\ldots\,\}, $$ i.e. $f_n$ is OEIS sequence A005250.

The paper arXiv:1709.05508 (Int. Math. Forum, vol.13, 2018, No.2, 65-78) conjectures that $f_n$ grows about as fast as a quadratic function of $n$; in fact computations suggest that $$ f_n \approx 0.25 n^2 + 0.5n; $$ this is equation $(8)$ in arXiv:1709.05508. If this quadratic trend continues as $n\to\infty$, then almost all positive even integers are not in $f_n$. That is, the sequence $t_n$ is infinite and contains almost all positive even integers: $$ \{t_n\}=\{10, 12, 16, 24, 26, 28, 30, 32, 38, 40, 42, 46, 48, 50, 54, 56, 58,60,\ldots\,\} $$ i.e. $t_n$ is OEIS sequence A063096.

Combining the above with the Polignac conjecture, we can also predict that almost all first-occurrence prime gaps are not maximal gaps; only a zero proportion of first-occurrence gaps are also maximal gaps. See T.R. Nicely's list of first-occurrence prime gaps, where maximal gaps are marked with an asterisk, and note that as we go down the list, asterisks become few and far between.