Gaps between nonconsecutive primes
In the last few days I dug my fangs into gaps between primes p[SUB]n[/SUB] and p[SUB]n+k[/SUB] (with k=1 these are the usual wellknown prime gaps, for k=2 see A[OEIS]144103[/OEIS], for k=3 see A[OEIS]339943[/OEIS], for k=4 see A[OEIS]339944[/OEIS]).
This can be seen as part of the effort to further improve the amount of empirical data related to prime gaps. Recently I found the paper [URL]https://arxiv.org/abs/2011.14210[/URL] (Abhimanyu Kumar, Anuraag Saxena: Insulated primes), which makes some predictions regarding k=2, but is based on quite limited empirical study. Here's a tidbit of data of especially large gaps for k=1..19 and p<6*10[SUP]12[/SUP]: [CODE] k CSG_max * p_n p_n+k 1 0.7975364 2614941710599 2614941711251 2 0.8304000 5061226833427 5061226834187 3 0.8585345 5396566668539 5396566669381 4 0.8729716 4974522893 4974523453 (largest CSG_max thus far) 5 0.8486459 137753857961 137753858707 6 0.8358987 5550170010173 5550170011159 7 0.8396098 3766107590057 3766107591083 8 0.8663070 11878096933 11878097723 9 0.8521843 1745499026867 1745499027983 10 0.8589305 5995661470529 5995661471797 11 0.8467931 5995661470481 5995661471797 12 0.8347906 5995661470529 5995661471893 13 0.8439277 5995661470529 5995661471977 14 0.8312816 5995661470481 5995661471977 15 0.7987377 5995661470471 5995661471977 16 0.7901341 5568288566663 5568288568217 17 0.7632862 396016668869 396016670261 18 0.7476038 396016668833 396016670261 19 0.7560424 968269822189 968269823761[/CODE]* A version of the CramÃ©rShanksGranville ratio. Only a quick spreadsheet formula, this could probably use some fine tuning[SUP]1)[/SUP], but for the time being, in this table [$]CSG = \Large \frac{gap}{(\log \frac{p_n+p_{n+k}}{2} +k1)^2}[/$] [SUP]1)[/SUP] I'd prefer something like M (the "merit") = Gram(p[SUB]n+k[/SUB])Gram(p[SUB]n[/SUB])k+1 where Gram(x) is Gram's version of Riemann's pi(x) approximation, and CSG = M[SUP]2[/SUP]/gap  pending negotiations... Calculations will have reached p ~ 7*10[SUP]12[/SUP] by tomorrow, and additionally for k=2 with p ~ 16*10[SUP]12[/SUP]. Not terribly fast, I admit. Does anybody know of any further work on this topic? 
For each k, what are the first few gaps with record CSG ratio? This is very interesting.

Greetings Bobby,
I'd like to run these numbers through Pari again before posting more inconsistent/approximate numbers. The formula with the term "k+1" (see [SUP]1)[/SUP] from previous post) is only working properly when CSG = max(0,M)[SUP]2[/SUP]/gap since M can be negative (because of the aforementioned term). Working out details like these takes me inordinately long... Good news is, for p = 8,281,634,108,677 and k = 19, I get a CSG > 1 with the roughandready version of the finetuned formula: gap = 1812, M = gap/log(p+gap/2)18 ~ 42.918 (there are 60.918 primes on average in a range of 1812 integers, i.e. 42.918 more than the 18 that are actually between the bounding primes), and CSG = M[SUP]2[/SUP]/gap ~ 1.0165. With p that large, there won't be much of a difference anymore when using Gram(x) in the calculation of CSG. 
When n=3, a big gap seems like 35617, 35671, 35677, 35729. There is a gap of 54 between 35617 and 35671, which is big for numbers of that size. After the gap of 6 between 35671 and 35677, there is another big gap of 52 between 35677 and 35729. Therefore, the 3gap between 35617 and 35729 is a surprisingly large prime gap.

[QUOTE=Bobby Jacobs;593106]When n=3, a big gap seems like 35617, 35671, 35677, 35729.[/QUOTE]
It doesn't only seem like a big gap, it's listed in A339943 as a(56), since 56=(3572935617)/2. I have a lot of data ready for submission, it just takes me longer to actually submit it, my schedule is pretty clogged at the moment... 
I looked at A115401, the record gaps between primes 3 apart, and it turns out that the gap of 112 between 35617 and 35729 is very big. The sequence starts out smoothly. After the initial 5, every even number from 8 to 36 is in the sequence. There are not many even numbers missing up to 68. Then, it jumps to 78, 84, and a really big leap to 112. That corresponds to the 35617, 35729 gap. It is an enormous prime gap!

I have uploaded some data for posterity, differences d between primes p[SUB]n[/SUB] and p[SUB]n+k[/SUB] for k <= 130 and d <= 740, see 2nd link for A[OEIS]086153[/OEIS].

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Some data in the attachment, just to show off.
The interested reader might also like to check, for instance, the differences p(n+42)p(n) for p in the range [327076775000..327076783000]. Makes for a nice graph. And this related result, 100 primes in the range p+[1..8349] while there are no primes in q+[1..8349], with q < p, still appears to be unmatched: [URL]https://www.mersenneforum.org/showpost.php?p=479832&postcount=86[/URL] Excuse my being a bit cocky today :wink: 
[QUOTE=mart_r;595755]Some data in the attachment, just to show off.
The interested reader might also like to check, for instance, the differences p(n+42)p(n) for p in the range [327076775000..327076783000]. Makes for a nice graph. And this related result, 100 primes in the range p+[1..8349] while there are no primes in q+[1..8349], with q < p, still appears to be unmatched: [URL]https://www.mersenneforum.org/showpost.php?p=479832&postcount=86[/URL] Excuse my being a bit cocky today :wink:[/QUOTE] Cocky! 
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[QUOTE=robert44444uk;595925]Cocky![/QUOTE]
What? What I wrote did look a little conceited to me:smile: [QUOTE=mart_r;595755]The interested reader might also like to check, for instance, the differences p(n+42)p(n) for p in the range [327076775000..327076783000]. Makes for a nice graph. [/QUOTE] That is, using a certain style of graph and a little imagination   V 
[QUOTE=mart_r;595952]What? What I wrote did look a little conceited to me:smile:
[/QUOTE] Nah, not really, Excellent work as always mart_r 
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