20201101, 11:36  #111 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2·3·7·137 Posts 
It strikes me that we are sieving the wrong dimension. In A*p#/d+x we currently sieve over x. Why don't we sieve over A? For each A we will count how many sieves over x it has survived. The As with the lowest score can then be looked at in the traditional way. Generally, we have a much larger range of A than the maximum gap size so the number of sieves should be reduced. This should mean deeper sieving.

20201103, 15:46  #112  
Jun 2003
Oxford, UK
3565_{8} Posts 
Quote:


20201104, 11:45  #113  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
2×3×7×137 Posts 
Quote:
Code:
0% 5% 10% 15% 20% 25% 30% 1.599538e10 4.355472e08 3.659155e07 4.189505e07 1.694673e06 2.834374e06 1.046412e05 35% 40% 45% 50% 55% 60% 65% 1.260456e05 1.339614e05 1.423743e05 1.597335e05 6.638588e05 1.672100e04 1.875974e04 70% 75% 80% 85% 90% 95% 100% 6.385621e04 9.265170e04 9.913896e04 1.035973e03 1.078905e03 1.138927e03 1.744451e03 Here are the results of a regression fit predicting the count remaining as the gcd of A and the divisor varies. 11 is the best by quite a bit and then 1, 7 and 77 are fairly close together. The fit is amazingly accurate. Code:
Call: lm(formula = count ~ .  1, data = sparse[sparse$k > 0, ]) Residuals: Min 1Q Median 3Q Max 137.862 17.733 0.148 17.603 143.204 Coefficients: Estimate Std. Error t value Pr(>t) gcddiv1 2.029e+03 5.882e02 34491 <2e16 *** gcddiv2 3.314e+03 5.882e02 56349 <2e16 *** gcddiv3 2.561e+03 8.318e02 30785 <2e16 *** gcddiv5 2.166e+03 1.176e01 18411 <2e16 *** gcddiv6 4.361e+03 8.318e02 52425 <2e16 *** gcddiv7 2.032e+03 1.441e01 14105 <2e16 *** gcddiv10 3.910e+03 1.176e01 33236 <2e16 *** gcddiv11 1.963e+03 1.860e01 10554 <2e16 *** gcddiv14 3.765e+03 1.441e01 26135 <2e16 *** gcddiv15 3.153e+03 1.664e01 18952 <2e16 *** gcddiv21 2.871e+03 2.038e01 14089 <2e16 *** gcddiv22 3.571e+03 1.860e01 19199 <2e16 *** gcddiv30 5.100e+03 1.664e01 30658 <2e16 *** gcddiv33 2.693e+03 2.630e01 10239 <2e16 *** gcddiv35 2.344e+03 2.882e01 8135 <2e16 *** gcddiv42 5.042e+03 2.038e01 24743 <2e16 *** gcddiv55 2.216e+03 3.720e01 5957 <2e16 *** gcddiv66 4.798e+03 2.631e01 18240 <2e16 *** gcddiv70 4.530e+03 2.882e01 15721 <2e16 *** gcddiv77 2.048e+03 4.556e01 4495 <2e16 *** gcddiv105 3.678e+03 4.075e01 9025 <2e16 *** gcddiv110 4.285e+03 3.720e01 11518 <2e16 *** gcddiv154 4.117e+03 4.556e01 9035 <2e16 *** gcddiv165 3.420e+03 5.261e01 6500 <2e16 *** gcddiv210 5.921e+03 4.075e01 14529 <2e16 *** gcddiv231 3.096e+03 6.443e01 4806 <2e16 *** gcddiv330 5.648e+03 5.260e01 10738 <2e16 *** gcddiv385 2.485e+03 9.111e01 2728 <2e16 *** gcddiv462 5.580e+03 6.443e01 8662 <2e16 *** gcddiv770 5.004e+03 9.111e01 5493 <2e16 *** gcddiv1155 4.044e+03 1.289e+00 3138 <2e16 *** gcddiv2310 6.571e+03 1.290e+00 5094 <2e16 ***  Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 26.81 on 999967 degrees of freedom Multiple Rsquared: 0.9999, Adjusted Rsquared: 0.9999 Fstatistic: 4.568e+08 on 32 and 999967 DF, pvalue: < 2.2e16 

20201104, 12:03  #114 
"Seth"
Apr 2019
3^{2}×23 Posts 
More on this today. I'm going to releasing my code which can do this interval is a handful of seconds.
Last fiddled with by SethTro on 20201104 at 12:04 
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