3-tuples and 4-tuples

MattcAnderson · 2020-12-31, 08:53

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
http://oeis.org/A022004
http://oeis.org/A022005

prime constellations with 4 numbers
http://oeis.org/A007530

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 from 1999 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
======================================================================

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 -
https://sites.google.com/site/primeconstellations/

Regards,
Matt

Batalov · 2020-12-31, 10:05

Even so, quad table in compressed form is already available in A014561 above 10,000 terms. (with "5" easily added in front).
Triplet tables will take less than a split second to compute.

It is like expanding A008585 to 10,000 terms.
Or A008596.
There are so many more sequences that can be extended!

MattcAnderson · 2020-12-31, 17:25

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
http://oeis.org/A213645
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: "more".

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

JeppeSN · 2021-01-18, 15:43

The official policy of OEIS disagrees.

On https://oeis.org/SubmitB.html 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.

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 where there are none.
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

2020-12-31, 08:53	#1
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 29E₁₆ Posts	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 http://oeis.org/A022004 http://oeis.org/A022005 prime constellations with 4 numbers http://oeis.org/A007530 Right now, the OEIS is judging my .b files. Wish me luck. Matt

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2020-12-31, 09:59	#3
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 2·5·67 Posts	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 - https://sites.google.com/site/primeconstellations/ Regards, Matt

2020-12-31, 10:05	#4
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2×13×359 Posts	Even so, quad table in compressed form is already available in A014561 above 10,000 terms. (with "5" easily added in front). Triplet tables will take less than a split second to compute. It is like expanding A008585 to 10,000 terms. Or A008596. There are so many more sequences that can be extended!

2020-12-31, 18:41	#6
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2×13×359 Posts	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: "more". A good rule of thumb: If a sequence doesn't have keyword "more" (or even more so, has "easy"!), then it doesn't need more terms!

2021-01-18, 17:01	#8
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2476₁₆ Posts	I don't see a contradiction here. This talks about "please send us b-files" meaning where there are none. 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".

2021-01-19, 22:31	#9
MattcAnderson "Matthew Anderson" Dec 2010 Oregon, USA 1010011110₂ Posts	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