20090328, 13:56  #1 
Dec 2008
you know...around...
2×7^{3} Posts 
Sum of reciprocals of prime ktuplets
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 ktuplets 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 20090328 at 14:03 
20090328, 15:06  #2  
Nov 2003
2^{2}·5·373 Posts 
Quote:
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 77tuples is very very hard. 

20090328, 15:43  #3  
Dec 2008
you know...around...
2·7^{3} Posts 
Quote:
http://anthony.d.forbes.googlepages....ets.htm#define Quote:
Whether one could give more precise results is open to discussion. Last fiddled with by mart_r on 20090328 at 15:45 

20090328, 16:26  #4  
Nov 2003
2^{2}·5·373 Posts 
Quote:
Can you show us the first two 77tuples that you used? Are we using the same definition? Please give us the exact definition you are using for a ktuple. a 77tuple is not just a set of 77 consecutive primes, for example. 23,29,31 are 3 consecutive primes, but they are not a 3tuple. A ktuple should have the MINIMAL span that occurs i.o. for 3tuples, that span is 6. 

20090328, 18:00  #5 
Dec 2008
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2·7^{3} 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 ktuplet 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 77tuplets with the initial primes being 41, 43, and 47, and a span of 420. Last fiddled with by mart_r on 20090328 at 18:01 
20090328, 21:25  #6  
Nov 2003
2^{2}·5·373 Posts 
Quote:
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) 

20090328, 22:30  #7 
Dec 2008
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2×7^{3} 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 77tuplet; 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 77tuplet less than googol. Or is it? 
20090328, 23:40  #8  
Nov 2003
2^{2}·5·373 Posts 
Quote:
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..... 

20090329, 07:57  #9 
Dec 2008
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2·7^{3} Posts 
Ah, I see. A tuplet refers to a certain constellation for one k, not every possible constellation.
My bad. But what is the collective name for all the possible tuplets for each k, then? If you give me a few days, I'll see what I can do. 
20090404, 16:10  #10 
Dec 2008
you know...around...
2AE_{16} 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) It's too sunny outside to complete the table right now, I'm going to finish it later. 
20090405, 07:29  #11 
Dec 2008
you know...around...
2×7^{3} 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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