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-   -   3-tuples and 4-tuples (https://www.mersenneforum.org/showthread.php?t=26359)

MattcAnderson 2020-12-31 08:53

3-tuples and 4-tuples
 
Hi all,

Prime Constellations are a special case of k-tuples. I had my computer do 3 calculations for k-tuples with patterns [0,2,6]; [0,4,6] and [0,2,6,8]
These are for The Online Encyclopedia of Integer Sequences.
I extend the tables from 1,000 to 10,000 numbers.

prime constellations with 3 numbers
[URL="http://oeis.org/A022004"]http://oeis.org/A022004[/URL]
[URL="http://oeis.org/A022005"]http://oeis.org/A022005[/URL]

prime constellations with 4 numbers
[URL="http://oeis.org/A007530"]http://oeis.org/A007530[/URL]

Right now, the OEIS is judging my .b files. Wish me luck.

Matt

Batalov 2020-12-31 09:16

But why?
Thomas Nicely's tables [B]from 1999[/B] had shown billions of quads.
Why extend it to 10000?
[CODE] Table 1.
Counts of prime quadruplets and estimates of Brun's B_4 constant.
======================================================================
x pi_4(x) delta_4(x) S_4(x) F_4(x)
======================================================================
10 1 10.29 0.510689310689311 0.964070321217938
100 2 11.60 0.789976586880612 0.846649213196690
1000 5 11.49 0.853473194253130 0.870265083531968
10000 12 12.17 0.863733192400183 0.870817270689693
10^05 38 14.88 0.867011003684134 0.870638051768363
10^06 166 17.68 0.868379532753497 0.870478518913352
10^07 899 -36.05 0.869267876960829 0.870589687487152
10^08 4768 -33.36 0.869705293632323 0.870590803418512
10^09 28388 8.84 0.869966856425087 0.870588778250229
10^10 180529 545.93 0.870134891176928 0.870588272187457
10^11 1209318 638.22 0.870247695545365 0.870588327409023
10^12 8398278 -3699.97 0.870326020813441 0.870588394083423
10^13 60070590 4848.36 0.870382016088034 0.870588379770569
10^14 441296836 -6103.68 0.870423153466140 0.870588379781931
2.0e+14 807947960 2717.36 0.870433368925933 0.870588379517423
3.0e+14 1151928827 -12660.14 0.870438957019776 0.870588379757893
4.0e+14 1482125418 -15032.60 0.870442759816539 0.870588379787802
5.0e+14 1802539207 -23557.26 0.870445621161320 0.870588379871401
6.0e+14 2115416076 -35177.17 0.870447903679533 0.870588379961704
7.0e+14 2422194981 -49882.89 0.870449795732922 0.870588380059497
8.0e+14 2723839871 -35301.69 0.870451407176393 0.870588379983029
9.0e+14 3021126140 -38141.52 0.870452807976233 0.870588379996686
1.0e+15 3314576487 -26197.22 0.870454044834374 0.870588379948604
1.1e+15 3604646822 -19485.07 0.870455150797010 0.870588379922608
1.2e+15 3891706125 -36034.00 0.870456149967533 0.870588379981021
1.3e+15 4175985018 -18182.67 0.870457060200978 0.870588379923299
1.4e+15 4457782901 -24552.75 0.870457895584737 0.870588379943262
1.5e+15 4737286827 -38254.45 0.870458666969910 0.870588379980055
1.6e+15 5014641832 -29496.94 0.870459383002448 0.870588379958289
======================================================================[/CODE]
Computing 10000 triplets or quads on the fly would be likely faster than fetching it from internet.

MattcAnderson 2020-12-31 09:59

Hi again,

Extending the tables from 1,000 to 10,000 potentially makes
it more useful for other people. It may be easy for you
to calculate these numbers, but it may be difficult for some people.

You may have seen my webpage on prime constellations -
[URL="https://sites.google.com/site/primeconstellations/"]https://sites.google.com/site/primeconstellations/[/URL]

Regards,
Matt

Batalov 2020-12-31 10:05

Even so, quad table in compressed form is already available in A[OEIS]014561[/OEIS] [U]above 10,000 terms[/U]. (with "5" easily added in front).
Triplet tables will take less than a split second to compute.

It is like expanding A[OEIS]008585[/OEIS] to 10,000 terms.
Or A[OEIS]008596[/OEIS].
There are so many more sequences that can be extended! :rolleyes:

MattcAnderson 2020-12-31 17:25

1 Attachment(s)
Attached is the simple minded code that I used to calculate 10,000 terms of the prime constellation with 4 primes.

Here is a link to a constellation with 12 primes in OEIS.org
[URL="http://oeis.org/A213645"]http://oeis.org/A213645[/URL]
I used more complicated code to calculate these as fast as possible.

As computing power increases and becomes cheaper, we will be able to extend these tables. This interesting mathematical trivia may be useful in the future.

Regards,
Matt

Batalov 2020-12-31 18:41

The computational power becomes cheaper.
The storage price doesn't, - it remains relatively flat.
What's more, storage price compounds, it is what OEIS will continue paying continuously year after year.

This is exactly why sequences that cost less than a second to compute should be kept virtual. Every sequence contains "PROG" section. -- A cooking recipe that can be run to make a cake. Or a pizza. Why do pizza shops make pizzas on order? Why don't they make 10,000 of them and store them? "What if someone needs 10,000 pizzas at once?"

Some sequences are like pizza. Having a 100 terms (with a stretch, ok, a 1000) is useful for the "search" function, but more is wasteful. Some sequences are like diamonds: they have e.g. 11 terms and computing the 12th will take a skilled person a month and an unskilled person forever. Those are of value and have a special keyword: "[C]more[/C]".

A good rule of thumb: If a sequence doesn't have keyword "[C]more[/C]" (or even more so, has "[C]easy[/C]"!), then it [I]doesn't need more[/I] terms!

JeppeSN 2021-01-18 15:43

The official policy of OEIS disagrees.

On [url]https://oeis.org/SubmitB.html[/url] you can read:

[QUOTE]Question: Is there a rule of thumb for which sequences we want b-files for?
Amswer: Pretty much any sequence - the b-files are always useful.

Question: I would think that sequences that are too simple (like the odd numbers) would be excluded.
Answer: No, even the odd numbers have a b-file (useful for plotting one sequence against another, for example).

Question: what size of b-file is acceptable?
Answer: 100, 1000, 10000, 20000 terms are usual. But remember that the numbers cannot have more than about 1000 digits, or some of the programs will fail.[/QUOTE]

So just because it is faster to calculate a sequence than to fetch it over the internet, does not mean you should not submit the b-file.

/JeppeSN

Batalov 2021-01-18 17:01

I don't see a contradiction here.

This talks about "please send us b-files" meaning [I]where there are none[/I].
It also says 100 terms is just fine to have. Useful for plotting. They said it, and I said it.

They didn't say, "just for vanity purposes, extend the b-file of odd numbers from 10,000 terms to 20,000 terms".

MattcAnderson 2021-01-19 22:31

For 3 tuples and 4 tuples we only need to consider divisibility by 2 and 3. We do not need to bother with divisibility by 5. So 3 tuples and 4 tuples can be all prime numbers if all the numbers are congruent to 1 or 5 modulo 6.


We say that x is relatively prime to 6 if and only if x is congruent to 1 or 5 mod 6.

Regards
Matt


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