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 2009-03-28, 13:56 #1 mart_r     Dec 2008 you know...around... 23×101 Posts Sum of reciprocals of prime k-tuplets Besides Thomas R. Nicely's site about the sum of reciprocals of prime twins, triplets, and quadruplets, I can find zilch on the web about these sums for quintuplets, sextuplets and so on. So I thought I give this table to the masses: Code: Sum of reciprocals of prime k-tuplets sorted by k sorted by sum k sum k sum 2 1.902160583 3 1.934964252 3 1.934964252 2 1.902160583 4 0.87058838 5 1.42195 5 1.42195 77 1.4089330681 6 0.52378 9 1.1390562 7 0.390933 51 0.9829565244 8 0.719295 13 0.9680305875 9 1.1390562 15 0.9651348367 10 0.46417945 17 0.9393735314 11 0.4854560409 4 0.87058838 12 0.5043239654 22 0.8324197024 13 0.9680305875 19 0.8095081121 14 0.5376665605 8 0.719295 15 0.9651348367 14 0.5376665605 16 0.4894659799 6 0.52378 17 0.9393735314 18 0.5141724005 18 0.5141724005 53 0.5138993952 19 0.8095081121 52 0.5104864260 20 0.3419910221 12 0.5043239654 21 0.3508405796 78 0.4936626231 22 0.8324197024 38 0.4935563225 23 0.4469504194 16 0.4894659799 24 0.0000000000 11 0.4854560409 25 0.3808416978 27 0.4769515376 10 0.46417945 27 0.4769515376 23 0.4469504194 33 0.4295598447 50 0.4437456028 38 0.4935563225 33 0.4295598447 50 0.4437456028 7 0.390933 51 0.9829565244 25 0.3808416978 52 0.5104864260 21 0.3508405796 53 0.5138993952 20 0.3419910221 77 1.4089330681 24 and all others 78 0.4936626231 0.0000000000 Last fiddled with by mart_r on 2009-03-28 at 14:03
2009-03-28, 15:06   #2
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

1D3416 Posts

Quote:
 Originally Posted by mart_r Besides Thomas R. Nicely's site about the sum of reciprocals of prime twins, triplets, and quadruplets, I can find zilch on the web about these sums for quintuplets, sextuplets and so on. So I thought I give this table to the masses: Code: Sum of reciprocals of prime k-tuplets sorted by k sorted by sum k sum k sum 2 1.902160583 3 1.934964252 3 1.934964252 2 1.902160583 4 0.87058838 5 1.42195 5 1.42195 77 1.4089330681 6 0.52378 9 1.1390562 7 0.390933 51 0.9829565244 8 0.719295 13 0.9680305875 9 1.1390562 15 0.9651348367 10 0.46417945 17 0.9393735314 11 0.4854560409 4 0.87058838 12 0.5043239654 22 0.8324197024 13 0.9680305875 19 0.8095081121 14 0.5376665605 8 0.719295 15 0.9651348367 14 0.5376665605 16 0.4894659799 6 0.52378 17 0.9393735314 18 0.5141724005 18 0.5141724005 53 0.5138993952 19 0.8095081121 52 0.5104864260 20 0.3419910221 12 0.5043239654 21 0.3508405796 78 0.4936626231 22 0.8324197024 38 0.4935563225 23 0.4469504194 16 0.4894659799 24 0.0000000000 11 0.4854560409 25 0.3808416978 27 0.4769515376 10 0.46417945 27 0.4769515376 23 0.4469504194 33 0.4295598447 50 0.4437456028 38 0.4935563225 33 0.4295598447 50 0.4437456028 7 0.390933 51 0.9829565244 25 0.3808416978 52 0.5104864260 21 0.3508405796 53 0.5138993952 20 0.3419910221 77 1.4089330681 24 and all others 78 0.4936626231 0.0000000000
This makes little sense. Consider, e.g. prime triplets. There is not
one sum, but two, since there are two types of triplets.
n, n+2, n+6, and n, n+4, n+6. Yet we only see one value
for the sum. Similarly, as k increases, so does the number of possible
different sums.......

And I find it very surprising that one can get such accuracy for the
larger values of k. Please show us exactly what was done to get the
number for k=77 (say). Just finding 77-tuples is very very hard.

2009-03-28, 15:43   #3
mart_r

Dec 2008
you know...around...

23·101 Posts

Quote:
 Originally Posted by R.D. Silverman This makes little sense. Consider, e.g. prime triplets. There is not one sum, but two, since there are two types of triplets. n, n+2, n+6, and n, n+4, n+6. Yet we only see one value for the sum. Similarly, as k increases, so does the number of possible different sums.......
I considered the general notion of prime k-tuplets.

Quote:
 Originally Posted by R.D. Silverman And I find it very surprising that one can get such accuracy for the larger values of k. Please show us exactly what was done to get the number for k=77 (say). Just finding 77-tuples is very very hard.
For the "large" cases of k, I only need to take the first known k-tuplets into account to provide the given accuracy of ten digits, as further k-tuplets don't significantly change the value (Hardy and Littlewood would agree:). I know it's not too elaborate, but it's about the best I could throw together in a few hours.
Whether one could give more precise results is open to discussion.

Last fiddled with by mart_r on 2009-03-28 at 15:45

2009-03-28, 16:26   #4
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

22·3·7·89 Posts

Quote:
 Originally Posted by mart_r I considered the general notion of prime k-tuplets. http://anthony.d.forbes.googlepages....ets.htm#define For the "large" cases of k, I only need to take the first known k-tuplets into account to provide the given accuracy of ten digits, as further k-tuplets don't significantly change the value (Hardy and Littlewood would agree:). I know it's not too elaborate, but it's about the best I could throw together in a few hours. Whether one could give more precise results is open to discussion.

Can you show us the first two 77-tuples that you used? Are we
using the same definition? Please give us the exact definition you
are using for a k-tuple. a 77-tuple is not just a set of 77 consecutive
primes, for example. 23,29,31 are 3 consecutive primes, but they are
not a 3-tuple. A k-tuple should have the MINIMAL span that occurs
i.o. for 3-tuples, that span is 6.

 2009-03-28, 18:00 #5 mart_r     Dec 2008 you know...around... 23×101 Posts Okay now, to quote the definition from Mr Forbes' page: We first define s(k) to be the smallest number s for which there exist k integers b1 < b2 < ... < bk, bk − b1 = s and, for every prime q, not all the residues modulo q are represented by b1, b2, ..., bk. A prime k-tuplet is then defined as a sequence of consecutive primes {p1, p2, ..., pk} such that for every prime q, not all the residues modulo q are represented by p1, p2, ..., pk, pk − p1 = s(k). Under this definition there are three admissible 77-tuplets with the initial primes being 41, 43, and 47, and a span of 420. Last fiddled with by mart_r on 2009-03-28 at 18:01
2009-03-28, 21:25   #6
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

22×3×7×89 Posts

Quote:
 Originally Posted by mart_r Okay now, to quote the definition from Mr Forbes' page: We first define s(k) to be the smallest number s for which there exist k integers b1 < b2 < ... < bk, bk − b1 = s and, for every prime q, not all the residues modulo q are represented by b1, b2, ..., bk. A prime k-tuplet is then defined as a sequence of consecutive primes {p1, p2, ..., pk} such that for every prime q, not all the residues modulo q are represented by p1, p2, ..., pk, pk − p1 = s(k). Under this definition there are three admissible 77-tuplets with the initial primes being 41, 43, and 47, and a span of 420.

Excellent. We agree on the defintion. So when you post a sum for k=77
which of the three tuplets are you summing? (and the same question
applies to the others)

 2009-03-28, 22:30 #7 mart_r     Dec 2008 you know...around... 14508 Posts Oy. You make it sound like I made a major mistake here. I proceed in the same way as one does when he calculates Brun's constant. That means, I take the sum over all primes of every 77-tuplet; i.e. 1/41+1/43+1/47+1/53+...+1/461 + 1/43+1/47+1/53+1/59+...+1/463 + 1/47+...+1/467 Now I don't think there's a very high probability that there's a fourth 77-tuplet less than googol. Or is it?
2009-03-28, 23:40   #8
R.D. Silverman

"Bob Silverman"
Nov 2003
North of Boston

22·3·7·89 Posts

Quote:
 Originally Posted by mart_r Oy. You make it sound like I made a major mistake here. I proceed in the same way as one does when he calculates Brun's constant. That means, I take the sum over all primes of every 77-tuplet; i.e. 1/41+1/43+1/47+1/53+...+1/461 + 1/43+1/47+1/53+1/59+...+1/463 + 1/47+...+1/467 Now I don't think there's a very high probability that there's a fourth 77-tuplet less than googol. Or is it?
There is nothing wrong.. We simply need clarification.

You are not distinguishing the tuplets when you sum then together.

For k = 3, you could sum over the n,n+2, n+6 tuplet or over
the n, n+4, n+6 tuplet separately.... They are distinct tuplets.
Instead you are summing over both together....You are not treating them
as different.

I'd be curious as to how the answer differs if you sum each one
sepaately.....

2009-03-29, 07:57   #9
mart_r

Dec 2008
you know...around...

23×101 Posts

Ah, I see. A tuplet refers to a certain constellation for one k, not every possible constellation.
But what is the collective name for all the possible tuplets for each k, then?

Quote:
 Originally Posted by R.D. Silverman I'd be curious as to how the answer differs if you sum each one separately.....
If you give me a few days, I'll see what I can do.

 2009-04-04, 16:10 #10 mart_r     Dec 2008 you know...around... 23×101 Posts Had a lot to do this week, but anyway... the table up to k=18: Code:  k sum tuplet form 2 1.902160583 p+0,2 3 1.097851039 p+0,2,6 3 0.837113212 p+0,4,6 4 0.870588380 p+0,2,6,8 5 0.94459 p+0,2,6,8,12 5 0.47736 p+0,4,6,10,12 6 0.52378 p+0,4,6,10,12,16 7 0.389559 p+0,2,6,8,12,18,20 7 0.001374 p+0,2,8,12,14,18,20 8 0.4165345 p+0,2,6,8,12,18,20,26 8 0.3026246 p+0,2,6,12,14,20,24,26 8 0.0001362 p+0,6,8,14,18,20,24,26 9 0.44092376 p+0,2,6,8,12,18,20,26,30 9 0.37334994 p+0,4,6,10,16,18,24,28,30 9 0.32467050 p+0,2,6,12,14,20,24,26,30 9 0.00011202 p+0,4,10,12,18,22,24,28,30 10 0.464179446 p+0,2,6,8,12,18,20,26,30,32 10 0.000000002 p+0,2,6,12,14,20,24,26,30,32 11 0.4854560408 p+0,2,6,8,12,18,20,26,30,32,36 12 0.5043239654 p+0,2,6,8,12,18,20,26,30,32,36,42 13 0.5212731179 p+0,2,6,8,12,18,20,26,30,32,36,42,48 13 0.4467574696 p+0,4,6,10,16,18,24,28,30,34,40,46,48 14 0.5376665605 p+0,2,6,8,12,18,20,26,30,32,36,42,48,50 15 0.5525919337 p+0,2,6,8,12,18,20,26,30,32,36,42,48,50,56 15 0.4125429030 p+0,2,6,12,14,20,24,26,30,36,42,44,50,54,56 16 0.4894659799 p+0,4,6,10,16,18,24,28,30,34,40,46,48,54,58,60 17 0.5021242078 p+0,4,6,10,16,18,24,28,30,34,40,46,48,54,58,60,66 17 0.4372493236 p+0,2,6,12,14,20,24,26,30,36,42,44,50,54,56,62,66 18 0.5141724005 p+0,4,6,10,16,18,24,28,30,34,40,46,48,54,58,60,66,70 (values for k=2, 3 and 4 from Thomas R. Nicely's page) Looks not very appealing to me, or how about you? It's too sunny outside to complete the table right now, I'm going to finish it later.
 2009-04-05, 07:29 #11 mart_r     Dec 2008 you know...around... 80810 Posts Code: 19 0.5254083556 p+0,4,6,10,16,18,24,28,30,34,40,46,48,54,58,60,66,70,76 19 0.2840997565 p+0,4,6,10,16,22,24,30,34,36,42,46,52,60,64,66,70,72,76 20 0.3419910221 p+0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80 21 0.3508405796 p+0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80,84 22 0.4381008619 p+0,4,10,12,18,22,24,28,34,40,42,48,52,54,60,64,70,78,82,84,88,90 22 0.3943188405 p+0,6,8,14,18,20,24,30,36,38,44,48,50,56,60,66,74,78,80,84,86,90 23 0.4469504194 p+0,4,10,12,18,22,24,28,34,40,42,48,52,54,60,64,70,78,82,84,88,90,94 25 0.3808416978 p+0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80,84,98,102,108,110 27 0.4769515376 p+0,4,10,12,18,22,24,28,34,40,42,48,52,54,60,64,70,78,82,84,88,90,94,108,112,118,120 33 0.4295598447 p+0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80,84,98,102,108,110,120,122,128,134,138,144,150,152 38 0.4935563225 p+0,6,8,14,18,20,24,30,36,38,44,48,50,56,60,66,74,78,80,84,86,90,104,108,114,116,126,128,134,140,144,150,156,158,168,170,174,176 50 0.4437456028 p+0,4,6,10,16,22,24,30,34,36,42,46,52,60,64,66,70,72,76,90,94,100,102,112,114,120,126,130,136,142,144,154,156,160,162,174,186,190,192,196,202,204,214,220,226,232,234,240,244,246 51 0.5069528571 p+0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80,84,98,102,108,110,120,122,128,134,138,144,150,152,162,164,168,170,182,194,198,200,204,210,212,222,228,234,240,242,248,252 51 0.4760036673 p+0,6,10,12,16,22,28,30,36,40,42,48,52,58,66,70,72,76,78,82,96,100,106,108,118,120,126,132,136,142,148,150,160,162,166,168,180,192,196,198,202,208,210,220,226,232,238,240,246,250,252 52 0.5104864260 p+0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80,84,98,102,108,110,120,122,128,134,138,144,150,152,162,164,168,170,182,194,198,200,204,210,212,222,228,234,240,242,248,252,254 53 0.5138993952 p+0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80,84,98,102,108,110,120,122,128,134,138,144,150,152,162,164,168,170,182,194,198,200,204,210,212,222,228,234,240,242,248,252,254,264 77 0.4915027959 p+0,2,6,12,18,20,26,30,32,38,42,48,56,60,62,66,68,72,86,90,96,98,108,110,116,122,126,132,138,140,150,152,156,158,170,182,186,188,192,198,200,210,216,222,228,230,236,240,242,252,266,270,272,276,290,296,306,308,312,318,326,332,338,342,348,356,360,368,378,380,390,392,398,402,408,416,420 77 0.4692723792 p+0,4,10,16,18,24,28,30,36,40,46,54,58,60,64,66,70,84,88,94,96,106,108,114,120,124,130,136,138,148,150,154,156,168,180,184,186,190,196,198,208,214,220,226,228,234,238,240,250,264,268,270,274,288,294,304,306,310,316,324,330,336,340,346,354,358,366,376,378,388,390,396,400,406,414,418,420 77 0.4481578929 p+0,6,12,14,20,24,26,32,36,42,50,54,56,60,62,66,80,84,90,92,102,104,110,116,120,126,132,134,144,146,150,152,164,176,180,182,186,192,194,204,210,216,222,224,230,234,236,246,260,264,266,270,284,290,300,302,306,312,320,326,332,336,342,350,354,362,372,374,384,386,392,396,402,410,414,416,420 78 0.4936626231 p+0,2,6,12,18,20,26,30,32,38,42,48,56,60,62,66,68,72,86,90,96,98,108,110,116,122,126,132,138,140,150,152,156,158,170,182,186,188,192,198,200,210,216,222,228,230,236,240,242,252,266,270,272,276,290,296,306,308,312,318,326,332,338,342,348,356,360,368,378,380,390,392,398,402,408,416,420,422 all others: 0.0000000000

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