20201231, 08:53  #1 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}·3^{2}·5 Posts 
3tuples and 4tuples
Hi all,
Prime Constellations are a special case of ktuples. I had my computer do 3 calculations for ktuples 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 
20201231, 09:16  #2 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·7·337 Posts 
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 ====================================================================== 
20201231, 09:59  #3 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}·3^{2}·5 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 
20201231, 10:05  #4 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×7×337 Posts 

20201231, 17:25  #5 
"Matthew Anderson"
Dec 2010
Oregon, USA
720_{10} Posts 
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 
20201231, 18:41  #6 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}·7·337 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! 
20210118, 15:43  #7  
"Jeppe"
Jan 2016
Denmark
10101000_{2} Posts 
The official policy of OEIS disagrees.
On https://oeis.org/SubmitB.html you can read: Quote:
/JeppeSN 

20210118, 17:01  #8 
"Serge"
Mar 2008
Phi(4,2^7658614+1)/2
2^{2}×7×337 Posts 
I don't see a contradiction here.
This talks about "please send us bfiles" 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 bfile of odd numbers from 10,000 terms to 20,000 terms". 
20210119, 22:31  #9 
"Matthew Anderson"
Dec 2010
Oregon, USA
2^{4}×3^{2}×5 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 
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