Sum of reciprocals of prime k-tuplets

mart_r · 2009-03-28, 13:56

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

R.D. Silverman · 2009-03-28, 15:06

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.

mart_r · 2009-03-28, 15:43

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.
http://anthony.d.forbes.googlepages....ets.htm#define

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.

R.D. Silverman · 2009-03-28, 16:26

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.

mart_r · 2009-03-28, 18:00

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.

R.D. Silverman · 2009-03-28, 21:25

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)

mart_r · 2009-03-28, 22:30

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?

R.D. Silverman · 2009-03-28, 23:40

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.....

mart_r · 2009-03-29, 07:57

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?

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.

mart_r · 2009-04-04, 16:10

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.

mart_r · 2009-04-05, 07:29

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

2009-03-28, 18:00	#5
mart_r Dec 2008 you know...around... 3³·29 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

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2009-03-28, 22:30	#7
mart_r Dec 2008 you know...around... 3³×29 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?