20221003, 12:45  #67 
Dec 2008
you know...around...
1101101000_{2} Posts 
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 p_{n} = 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 20221003 at 12:49 
20221016, 22:22  #68 
May 2018
286_{10} Posts 
You should get paid for all of the work you do in finding prime numbers.

20221017, 11:50  #69 
Dec 2008
you know...around...
2^{3}·109 Posts 
The replies that I get mean more to me than any amount of money (which is too tight to mention anyway ).

20221125, 20:57  #70 
Dec 2008
you know...around...
2^{3}·109 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 kgap 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 
20230105, 22:04  #71  
Dec 2008
you know...around...
2^{3}×109 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 p18) Quote:
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:
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 

20230108, 23:53  #72 
May 2018
436_{8} 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.

20230109, 19:18  #73  
Dec 2008
you know...around...
2^{3}×109 Posts 
I'm star walkin'
Quote:
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 followup 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. 

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