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-   -   Gaps between non-consecutive primes (https://www.mersenneforum.org/showthread.php?t=27301)

 mart_r 2022-10-03 12:45

[QUOTE=Bobby Jacobs;614723]You should get paid for this.[/QUOTE]

For that comment? Yeah, that was a tremendous outburst of creativity possibly worthy of a Pulitzer :wink:
But you lifted my mood, so I give a small update.
Prime scarcities with CSG > 1 are hard to find these days, but recently I got p[SUB]n[/SUB] = 205,465,264,987,331 which has
CSG = 1.0024950 for k = 927,
CSG = 1.0028949 for k = 941, and
CSG = 1.0063712 for k = 939.

 Bobby Jacobs 2022-10-16 22:22

You should get paid for all of the work you do in finding prime numbers.

 mart_r 2022-10-17 11:50

The replies that I get mean more to me than any amount of money (which is too tight to mention anyway :smile:).

 mart_r 2022-11-25 20:57

Eins Zwei Drei Vier Fünf Sechs Sieben Acht

Not a record in terms of CSG, but a close contender - and the largest p for which a CSG (or "scarcity/paucity ratio") > 1 is known:
A region with exceptionally few k+1 consecutive primes for k=274..380 (listed only for CSG > 1)

[CODE]k k-gap p start CSG
274 12914 309292876045019 1.00720962
275 12950 309292876045019 1.00580100
276 12996 309292876045253 1.00890285
277 13050 309292876045019 1.01561044
278 13110 309292876045019 1.02503096
279 13156 309292876045093 1.02813121
280 13230 309292876045019 1.04390467
281 13244 309292876045019 1.03252630
282 13300 309292876044949 1.04014280
283 13318 309292876044931 1.03061882
284 13342 309292876044907 1.02384129
285 13398 309292876044731 1.03143202
286 13438 309292876044811 1.03184383
287 13518 309292876044731 1.05023223
288 13542 309292876044707 1.04343738
289 13566 309292876044683 1.03667817
290 13580 309292876044683 1.02549729
291 13590 309292876044673 1.01261680
292 13620 309292876044629 1.00864965
293 13662 309292876044731 1.00999419
294 13700 309292876044731 1.00957620
295 13728 309292876044731 1.00476373
313 14448 309292876045019 1.01377017
314 14462 309292876045019 1.00309794
315 14518 309292876044949 1.01051652
316 14552 309292876045019 1.00847219
317 14604 309292876045019 1.01416841
318 14634 309292876045019 1.01041245
319 14674 309292876044949 1.01095647
320 14736 309292876044731 1.02093989
321 14760 309292876044707 1.01462007
322 14784 309292876044683 1.00833143
323 14840 309292876044731 1.01572021
324 14892 309292876044731 1.02140336
325 14922 309292876044731 1.01768328
326 14960 309292876044731 1.01738801
327 14984 309292876044707 1.01114162
328 15008 309292876044683 1.00492570
331 15134 309292876045019 1.00919910
332 15180 309292876045253 1.01231622
333 15220 309292876045213 1.01289750
334 15256 309292876045177 1.01179308
335 15340 309292876045093 1.03098513
336 15414 309292876045019 1.04600439
337 15432 309292876045019 1.03721965
338 15484 309292876044949 1.04288334
339 15502 309292876044931 1.03413607
340 15526 309292876044907 1.02796453
341 15582 309292876044851 1.03529999
342 15622 309292876044811 1.03588507
343 15702 309292876044731 1.05337889
344 15726 309292876044707 1.04718529
345 15750 309292876044683 1.04102057
346 15768 309292876044683 1.03236619
347 15778 309292876044673 1.02041740
348 15804 309292876044629 1.01519709
349 15832 309292876044601 1.01083020
350 15850 309292876044601 1.00234451
351 15906 309292876044527 1.00959606
352 15924 309292876044527 1.00113860
379 16980 309292876043453 1.00929898
380 16998 309292876043453 1.00114922
[/CODE]

 mart_r 2023-01-05 22:04

New year, new exceptional gaps

New exceptional gaps, for p = 343,408,238,858,639 each:
[CODE]k gap CSG
254 12174 1.0071454
255 12230 1.0147599
256 12248 1.0048780 (also for p-18)[/CODE]

[QUOTE=mart_r;613704]The "puzzle" with the geometric mean of consecutive gaps can be generalized: for n consecutive gaps the average ratio of the largest gap divided by the smallest gap appears to be as follows (rounded to three decimal places for n>=4):
[CODE] n r_gm
1 1
2 4
3 8
4 12.642
5 17.758
6 23.249
7 29.053
8 35.121
9 41.422
10 47.927[/CODE]
I'm afraid these numbers will give me headaches.
1.8 n (0.38 + log n) is an asymptotic facsimile for n<=1000, but we want more than this.
[/QUOTE]

Since those numbers are taken from the underlying distribution process, it may also apply to primes in any admissible residue class r mod q.
As a heuristical reality check, here's a sample of 30 consecutive primes congruent to 7 mod 1983:
[CODE]prime; mod 1983=7 gap ratio ratio ratio ratio
10^2023+24724407 2 gaps 3 gaps 4 gaps 5 gaps
10^2023+25533471 809064 max/min max/min max/min max/min
10^2023+32002017 6468546 7.99510
10^2023+47417859 15415842 2.38320 19.0539
10^2023+53751561 6333702 2.43394 2.43394 19.0539
10^2023+59791779 6040218 1.04859 2.55220 2.55220 19.0539
10^2023+95803059 36011280 5.96192 5.96192 5.96192 5.96192
10^2023+95870481 67422 534.118 534.118 534.118 534.118
10^2023+106816641 10946160 162.353 534.118 534.118 534.118
10^2023+110156013 3339372 3.27791 162.353 534.118 534.118
10^2023+116216061 6060048 1.81473 3.27791 162.353 534.118
10^2023+117687447 1471386 4.11860 4.11860 7.43935 162.353
10^2023+126531627 8844180 6.01078 6.01078 6.01078 7.43935
10^2023+141193929 14662302 1.65785 9.96496 9.96496 9.96496
10^2023+145504971 4311042 3.40110 3.40110 9.96496 9.96496
10^2023+146421117 916146 4.70563 16.0043 16.0043 16.0043
10^2023+147230181 809064 1.13235 5.32843 18.1225 18.1225
10^2023+151517427 4287246 5.29902 5.29902 5.32843 18.1225
10^2023+154829037 3311610 1.29461 5.29902 5.29902 5.32843
10^2023+161190501 6361464 1.92096 1.92096 7.86275 7.86275
10^2023+167615421 6424920 1.00998 1.94012 1.94012 7.94118
10^2023+171010317 3394896 1.89252 1.89252 1.94012 1.94012
10^2023+172049409 1039092 3.26718 6.18321 6.18321 6.18321
10^2023+172398417 349008 2.97727 9.72727 18.4091 18.4091
10^2023+174893031 2494614 7.14773 7.14773 9.72727 18.4091
10^2023+177062433 2169402 1.14991 7.14773 7.14773 9.72727
10^2023+177895293 832860 2.60476 2.99524 7.14773 7.14773
10^2023+182031831 4136538 4.96667 4.96667 4.96667 11.8523
10^2023+190796691 8764860 2.11889 10.5238 10.5238 10.5238
10^2023+197411979 6615288 1.32494 2.11889 10.5238 10.5238

geometric mean: 3.70023 7.86887 13.4903 20.6196[/CODE]

This can be generalized further to gaps between non-consecutive primes as well - either dependent on one another when running through consecutive primes (e.g. in the case k=2: 3-7, 5-11, 7-13, 11-17 etc.) or independent (by taking the differences at every other prime like 3-7, 7-13, 13-19, 19-29 etc.). Eventually these numbers appear for k=2:
[CODE] r_gm r_gm (geometric mean of ratio of maximal vs. minimal gap of n consecutive gaps between a prime and the k'th next prime (here k=2), limits as prime --> oo)
n dep. indep.
1 1 1
2 1.85 2.43
3 2.88 3.77
4 3.85 5.05
5 4.82 6.25
6 5.76 7.40
7 6.69 8.50
8 7.59 9.56
9 8.47 10.58
10 9.33 11.58
11 10.17 12.54
12 11.00 13.47
13 11.81 14.39[/CODE]

By now I think I've lost my audience for good... [COLOR="White"](not to mention my mind ;)[/COLOR]

 Bobby Jacobs 2023-01-08 23:53

You have not lost me. It is interesting that the first 3 numbers are integers:1, 4, 8. However, the next numbers are not integers. I wonder why that is.

 mart_r 2023-01-09 19:18

I'm star walkin'

1 Attachment(s)
[QUOTE=Bobby Jacobs;622006]You have not lost me. It is interesting that the first 3 numbers are integers:1, 4, 8. However, the next numbers are not integers. I wonder why that is.[/QUOTE]

You're truly my most loyal follower in this thread!
To be honest, I'm not entirely sure it's exactly 8 at n=3, rather 8.000000 ± 0.000007, ballpark. I still don't know how to pin the numbers down exactly.
I'm not so much wondering why the first few numbers are integers, there are a lot of sequences that start out with integers before fractional or even irrational numbers appear. What concerns me more is that I wasn't able to find a satisfying asymptotic approximation formula for larger n.

As a follow-up to [URL="https://www.mersenneforum.org/showpost.php?p=609511&postcount=55"]post # 55[/URL], I've attached the data for 2*10^14 < p < 3.33333333333333*10^14, k <= 109.
Meanwhile I've also looked for T(38,16) up to 2*10^16 - assuming the attached list in post # 58 is complete -, to no avail.

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