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2022-10-03, 12:45   #67
mart_r

Dec 2008
you know...around...

24×53 Posts

Quote:
 Originally Posted by Bobby Jacobs You should get paid for this.
For that comment? Yeah, that was a tremendous outburst of creativity possibly worthy of a Pulitzer
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 pn = 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.

Last fiddled with by mart_r on 2022-10-03 at 12:49

 2022-10-16, 22:22 #68 Bobby Jacobs     May 2018 23×5×7 Posts You should get paid for all of the work you do in finding prime numbers.
 2022-10-17, 11:50 #69 mart_r     Dec 2008 you know...around... 24×53 Posts The replies that I get mean more to me than any amount of money (which is too tight to mention anyway ).
 2022-11-25, 20:57 #70 mart_r     Dec 2008 you know...around... 24·53 Posts 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
2023-01-05, 22:04   #71
mart_r

Dec 2008
you know...around...

15208 Posts
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)

Quote:
 Originally Posted by mart_r 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 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.
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
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
By now I think I've lost my audience for good... (not to mention my mind ;)

 2023-01-08, 23:53 #72 Bobby Jacobs     May 2018 11816 Posts 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.
2023-01-09, 19:18   #73
mart_r

Dec 2008
you know...around...

24×53 Posts
I'm star walkin'

Quote:
 Originally Posted by Bobby Jacobs 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.
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 post # 55, 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.
Attached Files
 GNCP - CFC k=1..109 p=2e14..3.333e14.txt (90.5 KB, 1 views)

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