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Following on from finding the probability distribution for the 321 search http://www.primegrid.com/forum_thread.php?id=3781 I thought it would be interesting to find them for 27121.
Prime lists given by http://www.rieselprime.de/:
27*2^n-1: 1, 2, 4, 5, 8, 10, 14, 28, 37, 38, 70, 121, 122, 160, 170, 253, 329, 362, 454, 485, 500, 574, 892, 962, 1213, 1580, 2642, 2708, 4505, 8152, 11858, 13300, 15041, 16118, 16778, 19069, 22769, 29020, 30298, 30377, 35942, 42817, 42869, 62024, 74629, 90449, 91042, 117901, 128594, 143330, 152530, 157898, 175852, 282700, 340682, 444622, 627794, 729314, 777992, 1108214, 1163629, 1253870, 1902689, 3855094
27*2^n+1: 2, 4, 7, 16, 19, 20, 22, 26, 40, 44, 46, 47, 50, 56, 59, 64, 92, 175, 215, 275, 407, 455, 1076, 1090, 3080, 3322, 6419, 7639, 19360, 30500, 38770, 44164, 50696, 114760, 117784, 153847, 160874, 178739, 379880, 405514, 672007, 974752, 1164664, 1476347, 2218064
121*2^n-1: 1, 3, 21, 27, 37, 43, 91, 117, 141, 163, 373, 421, 1581, 2035, 10701, 18931, 21307, 51195, 64579, 156541, 302097, 334257, 368059, 383061, 410131, 494317, 541621, 990219, 1039965, 1526097, 1589157, 1695499, 1954243, 2033941, 4553899
121*2^n+1: 8, 12, 44, 84, 96, 228, 264, 320, 732, 788, 1808, 7928, 12264, 37784, 59196, 91112, 112208, 168716, 251184, 264632, 272052, 513020, 636564, 674360
By sorting the ratio of prime finds, graphing the distributions by giving each an equal weight and finding lines of best fit, got the following equations:
27*2^n-1: 1-16.777*e^(-2.832*X), where X = last prime * n
27*2^n+1: 1-9.399*e^(-2.225*X)
121*2^n-1: 1-3.538*e^(-1.242*X)
121*2^n+1: 1-3.327*e^(-1.241*X)
Running the numbers:
27*2^n-1, last prime 3,855,094
Range ; % chance ; % normalised to 4.1M
4.1 to 4.2M ; 5.85% ; 7.08%
4.2 to 4.3M ; 5.43% ; 6.58%
4.3 to 4.4M ; 5.05% ; 6.12%
4.4 to 4.5M ; 4.69% ; 5.68%
4.5 to 5.0M ; 18.91% ; 22.91%
5.0 to 5.5M ; 13.10% ; 15.87%
5.5 to 6.0M ; 9.07% ; 10.99%
6.0 to 6.5M ; 6.28% ; 7.61%
6.5 to 7.0M ; 4.35% ; 5.27%
7.0 to 7.5M ; 3.01% ; 3.65%
7.5 to 8.0M ; 2.09% ; 2.53%
8.0 to 8.5M ; 1.45% ; 1.75%
8.5 to 9.0M ; 1.00% ; 1.21%
9.0 to 9.5M ; 0.69% ; 0.84%
9.5 - 10.0M ; 0.48% ; 0.58%
10.0M to inf ; 1.08% ; 1.31%
27*2^n+1, last prime 2,218,064
Range ; % chance ; % normalised to 4.1M
4.1 to 4.2M ; 1.47% ; 9.54%
4.2 to 4.3M ; 1.33% ; 8.63%
4.3 to 4.4M ; 1.20% ; 7.81%
4.4 to 4.5M ; 1.09% ; 7.06%
4.5 to 5.0M ; 4.06% ; 26.41%
5.0 to 5.5M ; 2.46% ; 15.99%
5.5 to 6.0M ; 1.49% ; 9.68%
6.0 to 6.5M ; 0.90% ; 5.86%
6.5 to 7.0M ; 0.55% ; 3.55%
7.0 to 7.5M ; 0.33% ; 2.15%
7.5 to 8.0M ; 0.20% ; 1.30%
8.0 to 8.5M ; 0.12% ; 0.79%
8.5 to 9.0M ; 0.07% ; 0.48%
9.0 to 9.5M ; 0.04% ; 0.29%
9.5 - 10.0M ; 0.03% ; 0.18%
10.0M to inf ; 0.04% ; 0.27%
121*2^n-1, last prime 4,553,899
Range ; % chance ; % normalised to 5.26M
5.26 to 5.3M ; 0.80% ; 0.95%
5.3 to 5.4M ; 2.24% ; 2.67%
5.4 to 5.5M ; 2.18% ; 2.59%
5.5 to 6.0M ; 10.06% ; 11.96%
6.0 to 6.5M ; 8.78% ; 10.43%
6.5 to 7.0M ; 7.66% ; 9.10%
7.0 to 7.5M ; 6.68% ; 7.94%
7.5 to 8.0M ; 5.83% ; 6.93%
8.0 to 8.5M ; 5.09% ; 6.05%
8.5 to 9.0M ; 4.44% ; 5.28%
9.0 to 9.5M ; 3.87% ; 4.60%
9.5 - 10.0M ; 3.38% ; 4.02%
10 to 10.5M ; 2.95% ; 3.50%
10.5M to inf ; 20.18% ; 23.98%
121*2^n+1, last prime 674,360
Range ; % chance ; % normalised to 5.0M
5.0 to 5.1M ; 0.01% ; 16.80%
5.1 to 5.2M ; 0.00% ; 13.98%
5.2 to 5.3M ; 0.00% ; 11.63%
5.3 to 5.4M ; 0.00% ; 9.68%
5.4 to 5.5M ; 0.01% ; 8.05%
5.5 to 6.0M ; 0.00% ; 23.97%
6.0 to 6.5M ; 0.00% ; 9.55%
6.5 to 7.0M ; 0.00% ; 3.81%
7.0 to 7.5M ; 0.00% ; 1.52%
7.5 to 8.0M ; 0.00% ; 0.61%
8.0 to 8.5M ; 0.00% ; 0.24%
8.5 to 9.0M ; 0.00% ; 0.10%
9.0 to 9.5M ; 0.00% ; 0.04%
9.5 - 10.0M ; 0.00% ; 0.02%
10 to 10.5M ; 0.00% ; 0.01%
10.5M to inf ; 0.00% ; 0.00%
The last one needs special mention. Currently searching at 7.4x the last prime find. It is not unusual for a k to start misbehaving. What can once be a prime gusher can become relatively dry.
For example on the Riesel side, k <= 121 it is joining k = 7, 13, 23, 29, 37, 39, 43, 47, 51, 67, 71, 73, 79, 89, 95, 97, 101, 103, 109, 113, 119 in having such a high ratio.
Note: these are just statistics and do not refer to prime number theory.
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