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2022-08-01, 18:44
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#56 |
Dec 2008
you know...around...
15238 Posts |
Error terms, prime number scarcities, and trains
Just an intermediate result that made me go "hmmmm...". Suppose we assume \(CSG=1+O(1)\) for the gaps between non-consecutive primes, then, if I did the math right, this would imply that we also assume \(\pi(x)=Li(x)+O(\sqrt{x})\), i.e. the error term is smaller by a factor log x compared to the RH prediction. Correct [y/n]?
Code:
Outline from my train of thought: p_1 = 2 (or set p_0 = 0, say) p_k = x k = pi(x)-1 ~ pi(x) gap = x-2 ~ x m = Gram(x)-Gram(2)-k+1 ~ Gram(x)-pi(x) CSG = m*|m|/gap - but for simplicity suppose that m is positive (means we assume a scarcity instead of an abundance of primes; the error term works both ways anyway): CSG = m^2/gap ~ (Gram(x)-pi(x))^2/x CSG ~ 1 --> (Gram(x)-pi(x))^2 ~ x --> Gram(x)-pi(x) ~ sqrt(x) OTOH, if Gram(x)-pi(x) = O(sqrt(x)*log(x)), then CSG = m^2/gap ~ O((x*log²x)/x) ~ O(log²x) |
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2022-08-07, 19:05
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#57 |
May 2018
11816 Posts |
That is correct. By the way, you should submit the sequences in this thread to OEIS.
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2022-08-24, 17:03
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#58 | |
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Dec 2008
you know...around...
85110 Posts |
Quote:
I'd like to take the search for T(38,16) in A086153 up to 10^16, which will likely not be enough to find an example, but I still would like to see that case solved. It would take a bit more than a week with my program. I've identified 746 distinct constellations as shown in the attachment. I believe that list to be complete, albeit I'd be more content if that number was divisible by 4, so there's a slight possibility I have overlooked some constellations. If anyone with enough time on their hands feels inclined to do a quick double-check... For good measure, here's a batch of 79 instances where CSG > 1 for k > 1000: Code:
p k gap CSG 123146152018999 1152 44280 1.0322718 123146152018933 1154 44346 1.0310658 123146152018999 1127 43378 1.0263032 123146152018999 1126 43342 1.0260889 123146152018933 1129 43444 1.0250767 123146152018933 1156 44394 1.0249425 123146152018933 1128 43408 1.0248616 123146152018921 1155 44358 1.0247205 123146152018999 1157 44428 1.0246186 123146152018999 1133 43582 1.0242866 123146152018993 1153 44286 1.0242773 123146152018999 1138 43758 1.0242719 123146152018933 1159 44494 1.0234272 123146152018933 1135 43648 1.0230694 123146152018933 1140 43824 1.0230603 123146152018823 1158 44456 1.0226588 123146152019071 1151 44208 1.0221944 123146152018999 1125 43288 1.0209057 123146152018999 1139 43780 1.0206462 123146152018933 1141 43846 1.0194401 123146152018933 1160 44514 1.0192932 123146152018933 1130 43458 1.0192328 123146152019521 1113 42856 1.0183353 123146152019521 1112 42820 1.0181224 123146152018853 1132 43524 1.0180185 123146152018853 1131 43488 1.0178006 123146152018999 1094 42184 1.0176670 123146152018999 1091 42078 1.0176016 123146152018921 1136 43660 1.0166975 123146152019521 1143 43906 1.0165963 123146152018853 1162 44574 1.0164815 123146152018933 1096 42250 1.0164132 123146152018933 1093 42144 1.0163444 123146152019521 1119 43060 1.0163041 123146152019521 1124 43236 1.0162807 123146152018993 1134 43588 1.0162592 123146152018999 1095 42210 1.0150911 123146152018823 1163 44604 1.0150789 123146152018823 1144 43934 1.0146314 123146152019071 1137 43686 1.0141775 123146152018933 1097 42276 1.0138418 123146152019419 1142 43860 1.0136414 123146152018823 1161 44528 1.0135410 123146152019521 1111 42766 1.0129308 123146152019507 1114 42870 1.0124697 123146152018801 1164 44626 1.0115109 123146152019521 1116 42936 1.0112458 123146152018801 1145 43956 1.0110334 123146152019507 1120 43074 1.0104594 123146152019521 1080 41662 1.0097438 123146152019521 1077 41556 1.0096842 123146152018993 1092 42084 1.0094570 123146152019483 1115 42894 1.0093765 123146152018853 1099 42330 1.0092682 123146152018999 1088 41938 1.0080624 123146152018823 1100 42360 1.0078183 123146152019419 1117 42958 1.0076073 123146152018801 1146 43978 1.0074438 123146152018801 1165 44646 1.0074105 123146152018921 1098 42288 1.0073935 123146152019483 1121 43098 1.0073785 123146152019071 1090 42006 1.0073695 123146152019521 1081 41688 1.0071605 123146152019521 1079 41616 1.0067457 123146152019419 1147 44008 1.0060364 123146152019167 1150 44112 1.0056258 123146152019419 1123 43162 1.0056203 123146152018823 1101 42386 1.0052645 123146152019207 1149 44072 1.0043139 123146152018999 1089 41958 1.0038262 123146152019461 1122 43120 1.0037558 123146152019507 1078 41570 1.0037494 123146152019521 1118 42976 1.0028794 123146152018583 1167 44696 1.0019334 123146152019419 1148 44028 1.0019166 123146152018801 1102 42408 1.0016116 123146152019507 1082 41702 1.0012434 123146152019521 1074 41416 1.0001307 123146152018793 1166 44654 1.0000862 |
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2022-09-02, 18:14
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#59 | |
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Dec 2008
you know...around...
15238 Posts |
Dancing with tears in my eyes...
Quote:
They say that the gas pipeline Nord Stream 1 is kept shut indefinitely. So before power outrages become daily routine, I'd like to give an update on some numbers. - Gaps between non-consecutive primes, for k=104 to 1024 step 4: Code:
p <= 179133400000000 k gap CSG_max p 104 5656 1.0781126752 36683716323847 108 5824 1.0843154811 36683716323847 112 5940 1.0555010733 36683716323847 116 6052 1.0251605182 36683716323619 120 6220 1.0326995729 36683716323283 124 6388 1.0404373460 36683716323283 128 6510 1.0185881717 36683716323161 132 6642 1.0039006107 36683716323167 136 6742 0.9697674279 36683716323109 140 7292 0.9521920888 175478559288359 144 6840 0.9648817776 17674627574369 148 6992 0.9668891162 17674627574141 152 7126 0.9577979013 17674627574083 156 7460 0.9745283792 30512335335437 160 8144 0.9792679542 175478559288359 164 7732 0.9570891069 30512335335319 168 7946 0.9949905766 30512335334951 172 8100 0.9972610993 30512335334797 176 8254 0.9996701497 30512335334797 180 8364 0.9766164520 30512335335299 184 8510 0.9748349809 30512335335059 188 8736 1.0192157620 30512335334927 192 8892 1.0231919672 30512335334771 196 9004 1.0021590389 30512335334797 200 9148 1.0144816568 28330683392731 204 9324 1.0303925866 28330683392659 208 9492 1.0417729138 28330683392597 212 9630 1.0362667509 28330683392353 216 9778 1.0365415158 28330683392371 220 9856 0.9982886575 28330683392371 224 9974 0.9826932475 28330683392147 228 10058 0.9492927463 28330683392129 232 8294 0.9483514306 185067241757 236 9700 0.9564124562 5185992136441 240 9850 0.9639477964 5185992136453 244 10626 0.9420515461 28330683392597 248 10818 0.9664456553 28330683392371 252 10596 0.9477134052 12666866223047 256 11310 0.9390807333 52248744686339 260 11476 0.9481806463 52248744686197 264 11604 0.9386620116 52248744686069 268 11724 0.9254797708 52248744686197 272 11264 0.9300152206 12666866223047 276 12106 0.9116757369 68182243872601 280 11752 0.9125108382 21947823205027 284 11920 0.9253516441 21947823205027 288 12096 0.9421630582 21947823204943 292 12310 0.9417896525 25698372297889 296 12460 0.9453382500 25698372297691 300 12704 0.9950535482 25698372297029 304 12920 1.0312457318 25698372297029 308 13170 1.0847521505 25698372297029 312 13308 1.0819497629 25698372297029 316 13482 1.0972915901 25698372297029 320 13616 1.0925708301 25698372296963 324 13728 1.0770435304 25698372296873 328 13878 1.0804891085 25698372297029 332 13986 1.0633678090 25698372297007 336 14136 1.0669387810 25698372296963 340 14234 1.0453848066 25698372296873 344 15204 1.0287670437 127946496635897 348 15390 1.0445817969 127946496635761 352 15540 1.0446056952 127946496635611 356 15692 1.0455495832 127946496635459 360 15798 1.0266221073 127946496635459 364 15044 1.0256385680 25698372296963 368 15222 1.0426511752 25698372295019 372 15360 1.0409959251 25698372294839 376 15546 1.0617275885 25698372295033 380 15694 1.0647408322 25698372294457 384 15832 1.0631372016 25698372294409 388 15968 1.0606657649 25698372294611 392 16158 1.0831680032 25698372294421 396 16242 1.0568222056 25698372294337 400 16344 1.0390881483 25698372294563 404 16536 1.0622824463 25698372294457 408 16678 1.0627769150 25698372294421 412 16762 1.0372222537 25698372294337 416 16852 1.0147699971 25698372294457 420 16974 1.0067230990 25698372295033 424 17160 1.0268837758 25698372294421 428 17302 1.0276653950 25698372294421 432 17396 1.0075134540 25698372294611 436 17586 1.0293131103 25698372294421 440 17724 1.0284358125 25698372294409 444 17810 1.0051347652 25698372294323 448 17886 0.9779763169 25698372294253 452 17972 0.9554989984 25698372294281 456 18114 0.9566926179 25698372293557 460 18234 0.9487559428 25698372293809 464 18390 0.9558193895 25698372293597 468 18536 0.9587388800 25698372293597 472 19656 0.9486426201 112364701413971 476 19770 0.9862855261 93152147737279 480 19878 0.9719230760 93152147737199 484 19954 0.9455466698 93152147737237 488 19192 0.9433486203 25698372294421 492 20322 0.9755220092 93152147736727 496 20490 0.9844049639 93152147736559 500 20598 0.9704056982 93152147736451 504 20748 0.9724525235 93152147736301 508 20850 0.9564344564 93152147736199 512 21004 0.9600442813 93152147736073 516 21260 1.0021068423 93152147735789 520 21390 0.9965846355 93152147735659 524 21478 0.9753892159 93152147735599 528 21592 0.9641382100 93152147735371 532 21726 0.9603975747 93152147735351 536 21874 0.9618568153 93152147735203 540 21964 0.9421058879 93152147735113 544 22076 0.9305879709 93152147734973 548 22224 0.9321622171 93152147733739 552 22486 0.9751346521 93152147732647 556 22628 0.9744504782 93152147734421 560 22792 0.9818076325 93152147734171 564 22958 0.9898950736 93152147734091 568 23130 1.0001758329 93152147733919 572 23346 1.0266184470 93152147733703 576 23524 1.0391401441 93152147733553 580 23610 1.0178636352 93152147733467 584 23706 1.0005048976 93152147733553 588 23912 1.0230990399 93152147733137 592 24068 1.0275448938 93152147732981 596 24240 1.0378113630 93152147732723 600 24436 1.0568436976 93152147732641 604 24540 1.0423472371 93152147732509 608 24676 1.0395621760 93152147732401 612 24798 1.0317744784 93152147732251 616 24880 1.0098055650 93152147732197 620 25008 1.0043765193 93152147732069 624 25164 1.0088849117 93152147731913 628 25264 0.9936835657 93152147731813 632 25348 0.9731085473 93152147731729 636 25500 0.9762666351 93152147731549 640 25578 0.9539550727 93152147731499 644 25696 0.9455489291 93152147731381 648 25860 0.9528438164 93152147731217 652 26004 0.9533334554 93152147731073 656 26252 0.9893564159 93152147730797 660 26412 0.9952890412 93152147730637 664 26606 1.0129446324 93152147730443 668 26706 0.9982265121 93152147730371 672 26826 0.9904861287 93152147730223 676 26938 0.9801091903 93152147730139 680 27094 0.9846872883 93152147729983 684 27186 0.9677069785 93152147729891 688 27276 0.9502720219 93152147729983 692 27368 0.9337153701 93152147729891 696 27516 0.9356922025 93152147729561 700 27582 0.9109248623 93152147729467 704 27698 0.9026102400 93152147729561 708 27820 0.8962995612 93152147729143 712 27948 0.8919623539 93152147729143 716 28048 0.8787786737 93152147729143 720 27710 0.8927568473 54116590394771 724 27860 0.8963650091 54116590394621 728 27998 0.8960609520 54116590394483 732 28172 0.9075013936 54116590393157 736 28332 0.9143775739 54116590394149 740 28536 0.9357024342 54116590393991 744 28666 0.9327253951 54116590393861 748 28800 0.9310848576 54116590393777 752 28982 0.9451815024 54116590393499 756 29130 0.9481253683 54116590393447 760 29370 0.9814398420 54116590393157 764 29456 0.9638702316 54116590393121 768 29630 0.9753745155 54116590392947 772 29706 0.9546906758 54116590393157 776 29826 0.9485339103 54116590392947 780 29964 0.9482566835 54116590392929 784 30076 0.9395966369 54116590392451 788 30192 0.9322967009 54116590392947 792 30288 0.9186909882 54116590392289 796 30456 0.9280597127 54116590392121 800 30654 0.9470376796 54116590391873 804 30740 0.9302478398 54116590391837 808 31974 0.9117548600 159316577936029 812 30990 0.9217161811 54116590391873 816 31176 0.9367352659 54116590391351 820 32550 0.9554814455 159316577935453 824 31516 0.9567140006 54116590391011 828 31596 0.9382012070 54116590391011 832 31734 0.9380755356 54116590391113 836 31852 0.9316943869 54116590391011 840 31936 0.9148047257 54116590391077 844 32062 0.9110448935 54116590391077 848 32158 0.8981057702 54116590391011 852 32880 0.8916246643 93152147732647 856 33006 0.8873627713 93152147732641 860 32594 0.9048332804 54116590389887 864 32714 0.8993611673 54116590389863 868 34120 0.9116132357 159316577935453 872 34218 0.8986902090 159316577935453 876 34398 0.9093907279 159316577935453 880 34498 0.8971206895 159316577935453 884 34592 0.8832711645 159316577933411 888 34758 0.8899100180 159316577935453 892 34934 0.8993959518 159316577936297 896 35074 0.8986493801 159316577936233 900 35262 0.9115556093 159316577935969 904 35406 0.9119383256 159316577935453 908 35630 0.9351767250 159316577935601 912 35784 0.9384016321 159316577935453 916 35880 0.9250502187 159316577935453 920 36024 0.9254373323 159316577935433 924 36102 0.9071761352 159316577935601 928 36288 0.9194386246 159316577935453 932 36460 0.9277658577 159316577935453 936 36576 0.9202752667 159316577935601 940 36744 0.9274602593 159316577935453 944 36906 0.9329559987 159316577935453 948 37068 0.9384527900 159316577935453 952 37140 0.9186590345 159316577935453 956 37282 0.9185545605 159316577935969 960 37384 0.9073477992 159316577935969 964 37650 0.9418370777 159316577935601 968 37800 0.9439561470 159316577935451 972 37914 0.9360034504 159316577935423 976 38014 0.9242274696 159316577935969 980 37812 0.9276107833 120293264372867 984 38382 0.9474156903 159316577935601 988 38538 0.9512197726 159316577935453 992 38680 0.9511189880 159316577935453 996 38796 0.9438031196 159316577935453 1000 38866 0.9238826098 159316577935453 1004 39040 0.9326313213 159316577935453 1008 39126 0.9172648307 159316577935453 1012 39348 0.9391390323 159316577935601 1016 39538 0.9523097203 159316577935453 1020 39680 0.9522520689 159316577935601 1024 39856 0.9615749322 159316577935453 Code:
p <= 23388300000000 q gap CSG (conv.) p 4568 1548552 0.8572222356 1677084447851 4570 1183630 0.8390238160 1196563621633 4572 676656 0.9313083686 3312086153 4574 864486 0.8348603294 1749438037 4576 617760 0.8078001685 464384941 4578 691278 0.8429669902 84048460189 4580 1108360 0.8026462441 890252180611 4582 1383764 0.8499365180 720395477939 4584 1141416 0.7964331851 21652890442697 4586 1371214 0.8127600152 606453831427 4588 1601212 0.8876812908 3550398242161 4590 867510 0.8724168546 5747636061659 4592 1428112 0.8464431695 7482931558789 4594 1745720 0.8491640934 9894775021751 4596 896220 0.8192055813 417572047247 4598 1549526 0.8296428911 21799960507001 4600 1283400 0.7977381722 13502858057147 4602 1090674 0.8445938334 16897337246939 4604 1749520 0.8362905821 12528155746207 4606 428358 0.8045087092 15779549 4608 926208 0.8881770694 207016317479 4610 1212430 0.8526133843 1183984521847 4612 894728 0.9475156489 618769103 4614 461400 0.8323251172 177577073 4616 1583288 0.8432163277 2500655788181 4618 1454670 0.8260526488 991355520937 4620 328020 0.8965801195 300426827 4622 1687030 0.8202913337 9089241698347 4624 550256 0.8642679215 26284901 4626 1050102 0.8210523101 3403774701511 4628 1596660 0.8144418081 17046531702059 4630 1361220 0.7957535391 16338258362707 4632 291816 0.8541090988 2708653 4634 1181670 0.8315601474 431161340839 4636 1070916 0.7908290917 74835258193 4638 834840 0.8481347282 92370239231 4640 389760 0.8488446783 8560609 4642 1425094 0.8402156795 2200086910369 4644 1114560 0.8332081592 8272678130681 4646 1681852 0.8229584664 17244249871939 4648 957488 0.8394410220 28539747311 4650 916050 0.8669161631 7715831183377 4652 907140 0.8022040760 3800111563 4654 837720 0.9111719565 1022854519 4656 1164000 0.8197000840 16031665116419 4658 1481244 0.7793115567 6846908799389 4660 1337420 0.8169553941 7910942491217 4662 731934 0.7990676643 351440070953 4664 1455168 0.7900947961 8378888283431 4666 1843070 0.8398134556 21034084207667 4668 606840 0.7997013450 4009021879 4670 448320 0.8325428315 23630069 4672 1191360 0.8690884709 39202962193 4674 1051650 0.8288516679 7787632946129 4676 827652 0.8363384979 4784767627 4678 1468892 0.8470457965 672647311367 4680 336960 0.8186005024 161618917 4682 435426 0.8382468056 2520173 4684 1311520 0.7852421447 400661984803 4686 1077780 0.8318802396 16277427487337 4688 1392336 0.8603480773 269749617047 4690 1069320 0.8882226297 939523507907 4692 1102620 0.8986838410 6608920918927 4694 1825966 0.9167594012 4511534178661 4696 723184 0.8224917628 257098579 4698 361746 0.8481017107 19332767 4700 1207900 0.8281666227 1687877098657 4702 893380 0.8660946502 1254661021 4704 940800 0.7810442832 10036010128121 4706 1369446 0.8135118655 1330625650261 4708 630872 0.8454641507 139903367 4710 527520 0.8361774196 5812980889 4712 1526688 0.8205824387 5570936685233 4714 900374 0.8147104625 2546122913 4716 429156 0.7783377852 145721231 4718 1387092 0.8456133333 2444163196961 4720 1368800 0.8525735781 5931853116047 4722 798018 0.8208499645 63126509689 4724 1927392 0.9137557146 9631507165417 4726 1436704 0.8474484525 1081554964909 4728 997608 0.8826331534 457159963609 4730 1177770 0.7925444628 8252709146287 4732 723996 0.8275560049 2446077173 4734 1013076 0.7972189748 2227503505511 4736 374144 0.8089251738 1049177 4738 1331378 0.8449824963 322102867019 4740 616200 0.8374867911 35079078883 4742 1019530 0.8251804802 8239994179 4744 1892856 0.8523019576 19958061839741 4746 322728 0.8242463704 25539551 4748 356100 0.8672257307 161611 4750 1268250 0.8645165155 2502352535699 4752 1092960 0.8514531711 9258938335169 4754 1231286 0.7892730195 134353071551 4756 1112904 0.8082974840 58507371397 4758 494832 0.8244859283 734454311 4760 985320 0.7894969010 2395919782511 4762 1219072 0.9308036050 15409579657 4764 452580 0.8622914404 79985923 4766 1572780 0.8311913693 1739547200591 4768 1587744 0.8131646687 2956675908829 4770 887220 0.7789200810 13193937699571 4772 691940 0.8098206509 165985607 4774 1260336 0.8112539172 5739683343047 4776 448944 0.8442629551 90180059 4778 501690 0.9497720783 2376683 4780 1104180 0.8132485059 395693430791 4782 526020 0.8161806515 546710977 4784 1368224 0.8174372120 1683148638257 4786 789690 0.8598555091 322654259 4788 995904 0.8147188358 21770871850033 4790 1216660 0.8416658242 873768458473 4792 1269880 0.8330024589 92005688941 4794 661572 0.7960364440 20861219971 4796 1251756 0.8307730973 295346729533 4798 1679300 0.8446785055 3197716234463 4800 969600 0.8209328289 15571822465201 4802 388962 0.8869542712 1796987 4804 1710224 0.7782874527 13842179098073 4806 1206306 0.8333516049 13448938011913 4808 533688 0.8537185219 9679121 4810 274170 0.8519340567 570697 4812 307968 0.8415538293 3390703 4814 1473084 0.8271424430 1245798692843 4816 1247344 0.7877295603 1484176131853 4818 876876 0.8215487463 666432600907 4820 935080 0.8313203473 32487008371 4822 1750386 0.8577540614 4340143171271 4824 1201176 0.8698188410 6655443895757 4826 482600 0.8212477031 9305063 4828 1670488 0.7931469867 20747292365671 4830 454020 0.8785153116 4050952519 4832 1797504 0.8485095387 7995365520743 4834 1672564 0.8896834577 1301997325459 4836 614172 0.8180428160 8250971321 4838 1465914 0.7713956943 2688844111963 4840 1316480 0.8550815866 6997095903977 4842 1074924 0.8840456566 875855956663 4844 465024 0.8806644242 8374607 4846 688132 0.8601739780 77467669 4848 1110192 0.7856487086 8063436375157 4850 388000 0.8444615696 4839613 4852 1572048 0.8412827663 1143215029679 4854 961092 0.7804868212 973772706631 4856 1592768 0.8380868777 1446945776483 4858 1462258 0.8207158595 5283274725721 4860 704700 0.7812851332 286560716669 4862 1604460 0.8991146486 17381205974537 4864 812288 0.8810163626 486408491 4866 1124046 0.8356057290 3270645941561 4868 1630780 0.8590928041 1359358668541 4870 1344120 0.8735751658 1652489415169 4872 954912 0.8590073654 3091414128029 4874 731100 0.9439063939 54741157 4876 1482304 0.8250557169 1478412698227 4878 1034136 0.8224074757 1257853715861 4880 683200 0.7939954643 1562016139 4882 1635470 0.8558980255 1423810702421 4884 434676 0.8333452034 183889271 4886 747558 0.8734681702 619568773 4888 821184 0.7856745434 2810773463 4890 322740 0.9302432611 12434311 4892 826748 0.8516229144 451642253 4894 1747158 0.8243990452 6075857766151 4896 563040 0.8478787257 1071146539 4898 1077560 0.8095032996 22817244383 4900 764400 0.8228894964 16299646609 4902 676476 0.7730046246 28069830563 4904 818968 0.8186527675 600834991 4906 1501236 0.8367576017 2219095818727 4908 1173012 0.7979230970 10827220170911 4910 574470 0.8515239444 113538983 4912 1792880 0.8327591122 7574725685297 4914 545454 0.8596639821 4067682527 4916 1042192 0.8689515451 3954781777 4918 1091796 0.7998596761 17149465633 4920 319800 0.7628511092 72051739 4922 1717778 0.7963065272 16173410263259 4924 797688 0.7994122308 557381641 4926 1054164 0.8376718240 1072468422437 4928 1232000 0.8213657477 1376131327733 4930 1296590 0.8304159408 6598307178041 4932 1203408 0.7907077203 18298578679613 4934 1845316 0.7993224339 19412314489657 4936 1584456 0.8193829451 1466754802973 4938 1338198 0.8687111129 19679180418991 4940 350740 0.8672492232 4055353 4942 1225616 0.8124712787 404340010181 4944 726768 0.8416485436 9767120689 4946 1068336 0.8805527859 4180667681 4948 1603152 0.8345031758 1279124270959 4950 747450 0.8071641250 1159704007081 4952 891360 0.8149740548 1364101967 4954 1733900 0.8393620485 3501849078881 4956 594720 0.7723423113 16385484361 4958 1388240 0.9618399154 50569228469 4960 813440 0.8680593904 3930037487 4962 630174 0.8888727624 992107469 4964 1454452 0.8388185803 820409883907 4966 1455038 0.8268943841 1161574071641 4968 298080 0.8742764448 2054821 4970 685860 0.8377735104 3863252677 4972 1014288 0.8786013888 7231270463 4974 1218630 0.8120592137 11847620254121 4976 1771456 0.8635599699 3093357916003 4978 1762212 0.8347191886 11084980562807 4980 806760 0.7651954247 2047787890429 4982 1509546 0.8050403273 1443995177347 4984 1046640 0.8968469262 16174195679 4986 917424 0.7982685913 276068375017 4988 1281916 0.8583537673 87830579833 4990 1347300 0.7979239899 4407482390587 4992 524160 0.8022670052 905148287 4994 1812822 0.8971218655 9687146951987 4996 1533772 0.8372621149 582820048477 4998 324870 0.8595695977 18853409 5000 1365000 0.8722366178 1407287251891 5002 750300 0.8398345295 238656883 5004 415332 0.8897546225 19150451 |
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2022-09-17, 18:29
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#61 |
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May 2018
23·5·7 Posts |
No. I have never heard of that show.
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2022-09-18, 12:59
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#62 | ||
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Dec 2008
you know...around...
23·37 Posts |
Quote:
Using a Poisson distributed random model, i.e. with random variables 0 < xn < 1 turned into a function equivalent to the merit mn = -log(1-xn), we want two consecutive values m1 and m2 such that m1 >= m2 and then the geometric mean rgm of (m1, m2). For given x2, we use a function f(x2) that gives the geometric mean of m1/m2 for all m1 >= m2. We have \(\log (f(x_2)) = \frac{\int_{x_2}^1 \log(-\log(1-y)) \: \text{d}y}{1-x_2} - \log(-\log(1-x_2)).\) Taking all x2 into account, we would get \(\log(r_{gm})=2 \cdot \int_0^{1-\varepsilon} (1-z) \log(f(z)) \: \text{d}z\) and lim rgm = 4 for \(\varepsilon \to\) 0 - numerically. I don't yet know how to prove that mathematically, but as I said, I'm sure the tools are available and I leave that as an exercise for those who are more comfortable working with integrals as I am. Quote:
I don't use streaming services, but I would be curious whether it's available on any of the popular platforms, and unabridged at that. I know that it's at least partially censored in some countries... but I don't want to spoiler anything  Just, if you do, be careful to watch it in chronological order since it follows a single story plot.
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2022-09-19, 16:29
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#63 |
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Dec 2008
you know...around...
15238 Posts |
Where's the revolution? C'mon, people, you're letting me down!
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 12.641 5 17.757 6 23.249 7 29.052 8 35.121 9 41.423 10 47.928 1.8 n (0.38 + log n) is an asymptotic facsimile for n<=1000, but we want more than this. What is it that I should ask myself? |
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2022-09-21, 21:52
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#64 |
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Dec 2008
you know...around...
23·37 Posts |
r_gm @ n=3 = 12.64200 ± 0.000015
All work and no pay makes me wish life wouldn't be so dull.
Here, I give you the first occurrence gaps for k=1000 up to p=2*10^14. You'll find me in the kitchen. PS: To this day I've never expressed my continual deep appreciation for the brilliantly derived joke that came via that calculation from this old idea of mine. |
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2022-09-24, 11:57
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#65 | |
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Dec 2008
you know...around...
23×37 Posts |
I stand corrected
Quote:
While I'm at it, here are the values up to n=13, corrected to be within +/- about 2 sigma: Code:
1 1 2 4 3 8 4 12.642007 ± 0.000009 5 17.75797 ± 0.00002 6 23.24943 ± 0.00003 7 29.05309 ± 0.00003 8 35.12109 ± 0.00004 9 41.42190 ± 0.00004 10 47.92674 ± 0.00005 11 54.62285 ± 0.00005 12 61.47478 ± 0.00006 13 68.49414 ± 0.00006 |
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2022-10-01, 22:51
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#66 | |
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May 2018
23×5×7 Posts |
Quote:
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