mersenneforum.org Some transition probabilities
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 2012-03-15, 14:29 #1 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 22·32·179 Posts Some transition probabilities A.B.C.D means 2^A*3^B*5^C*7^D These are from 1250 lines from the 2^2*3 project Code: 2.0.0.0 0.890 2.0.0.0 0.044 2.0.1.0 *0.015 3.0.0.0* *0.015 1.0.0.0* 0.011 2.0.2.0 0.007 2.1.0.0 *0.004 7.0.1.0* 0.004 2.0.7.0 *0.004 4.0.0.0* *0.004 3.2.0.0* 0.004 2.0.3.0 2.1.0.0 0.832 2.1.0.0 0.052 2.0.0.0 0.047 2.1.1.0 0.044 2.2.0.0 0.010 2.3.0.0 0.008 2.1.2.0 0.003 2.0.1.0 0.003 2.2.1.0 0.003 2.4.0.0 2.1.1.0 0.608 2.1.1.0 0.278 2.1.0.0 0.089 2.1.2.0 0.013 2.1.3.0 0.013 2.2.1.0 2.2.0.0 0.620 2.2.0.0 0.092 2.1.0.0 0.063 2.2.1.0 0.049 2.3.0.0 0.042 2.0.0.0 0.021 2.2.2.0 0.021 2.1.1.0 0.021 2.5.0.0 0.014 2.2.3.0 *0.007 1.0.0.0* 0.007 2.0.1.0 0.007 2.3.1.0 0.007 2.4.0.0 *0.007 3.0.0.0* 0.007 2.5.1.0 *0.007 4.4.0.0* 0.007 2.6.0.0 Read this as 'if you have currently a factor 2^2*3^2 (and are about 115 digits), probability is about 0.092 that you will be 2^2*3 next iteration'. The probabilities are taken from at least 50 (about 300 for 2.1.0.0) lines. Ones with stars are where the power of two changed. Last fiddled with by fivemack on 2012-03-15 at 14:31
 2012-03-15, 16:08 #2 firejuggler     "Vincent" Apr 2010 Over the rainbow 2×32×149 Posts what about those marked with an asterisk? not enough data to be confident?
 2012-03-15, 16:57 #3 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 22×32×179 Posts Ones with stars are where the power of two changed.
 2012-03-15, 18:11 #4 firejuggler     "Vincent" Apr 2010 Over the rainbow 2·32·149 Posts sorry i prefer my data sorted vertically Code:  driver/guide next iteration % chance 2^2 2^2 0,89 2^2*5 0,044 2^3 0,015 2^2 0,015 2^2*5^2 0,011 2^2*3 0,007 2^7*5 0,004 2^2*5^7 0,004 2^4 0,004 2^3*3^2 0,004 2^2*5^3 0,004 2^2*3 2^2*3 0,832 2^2 0,052 2^2*3*5 0,047 2^2*3^2 0,044 2^2*3^3 0,01 2^2*3*5^2 0,008 2^2*5 0,003 2^2*3^2*5 0,003 2^2*3^4 0,003 2^2*3*5 2^2*3*5 0,608 2^2*3 0,278 2^2*3*5 0,089 2^2*3*5^3 0,013 2^2*3^2*5 0,013 2^2*3^2 2^2*3^2 0,62 2^2*3 0,092 2^2*3^2*5 0,063 2^2*3^3 0,049 2^2 0,042 2^2*3^2*5^2 0,021 2^2*3*5 0,021 2^2*3^5 0,021 2^2*3^2*5^3 0,014 2 0,007 2^2*5 0,007 2^2*3^3*5 0,007 2^2*3^4 0,007 2^3 0,007 2^2*3^5*5 0,007 2^4*3^4 0,007 2^2*3^6 0,007 another question, no 2^2*7 driver? Last fiddled with by firejuggler on 2012-03-15 at 18:12
2012-03-15, 23:28   #5
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

173116 Posts

Quote:
 Originally Posted by firejuggler another question, no 2^2*7 driver?
You can't get to 2^2*7 directly through 2^2 because sigma(2^2)=7. You have to go though another power of two.

 2012-03-15, 23:35 #6 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 11001001011002 Posts OK, here's the analysis for all the 110-to-120 digit numbers in the aliquot sequences I have run. Hope it's vaguely interesting to someone. Code: 1.0.0.0 (345) 0.841 1.0.0.0 0.070 1.0.0.1 0.041 1.0.1.0 0.014 1.0.0.2 0.009 1.0.1.1 *0.006 3.0.0.0* *0.006 2.0.0.0* 0.006 1.0.2.0 0.003 1.0.4.0 0.003 1.0.3.0 *0.003 2.0.2.0* 1.0.1.0 (73) 0.630 1.0.1.0 0.247 1.0.0.0 0.055 1.0.2.0 0.041 1.0.1.1 0.014 1.0.1.2 0.014 1.0.0.1 1.1.0.0 (237) 0.844 1.1.0.0 0.038 1.2.0.0 0.038 1.1.0.1 0.034 1.1.1.0 0.017 1.3.0.0 0.008 1.1.2.0 0.008 1.1.0.2 0.004 1.2.3.0 0.004 1.6.0.0 0.004 1.2.0.1 1.2.0.0 (172) 0.773 1.2.0.0 0.058 1.2.1.0 0.041 1.1.0.0 0.035 1.3.0.0 0.035 1.2.0.1 0.023 1.4.0.0 0.006 1.1.2.0 *0.006 2.1.0.0* 0.006 1.1.1.0 0.006 1.2.2.0 0.006 1.3.2.0 *0.006 4.1.1.0* 1.3.0.0 (52) 0.635 1.3.0.0 0.135 1.2.0.0 0.077 1.4.0.0 0.077 1.1.0.0 0.038 1.3.0.1 0.038 1.5.0.0 2.0.0.0 (740) 0.882 2.0.0.0 0.046 2.0.1.0 0.018 2.1.0.0 *0.008 1.0.0.0* *0.008 3.0.0.0* *0.005 4.0.0.0* 0.005 2.0.2.0 0.004 2.0.3.0 *0.003 3.2.0.0* 0.003 2.2.0.0 0.003 2.1.1.0 *0.001 4.0.1.0* *0.001 7.0.0.0* *0.001 3.0.1.0* 0.001 2.3.1.0 *0.001 6.0.0.0* *0.001 5.0.1.0* *0.001 7.0.1.0* 0.001 2.0.7.0 *0.001 7.1.0.0* 0.001 2.3.0.0 *0.001 1.0.3.0* 2.0.0.1 (98) 0.847 2.0.0.1 0.082 2.0.1.1 0.051 2.0.0.2 0.010 2.0.3.1 0.010 2.0.1.2 2.0.1.0 (121) 0.628 2.0.1.0 0.306 2.0.0.0 0.058 2.0.2.0 *0.008 3.0.0.0* 2.1.0.0 (558) 0.828 2.1.0.0 0.057 2.0.0.0 0.050 2.2.0.0 0.034 2.1.1.0 0.009 2.3.0.0 0.009 2.1.2.0 0.005 2.4.0.0 0.004 2.0.1.0 0.004 2.2.1.0 2.1.1.0 (95) 0.600 2.1.1.0 0.295 2.1.0.0 0.084 2.1.2.0 0.011 2.1.3.0 0.011 2.2.1.0 2.2.0.0 (203) 0.635 2.2.0.0 0.084 2.1.0.0 0.059 2.2.1.0 0.039 2.0.0.0 0.039 2.3.0.0 0.025 2.2.2.0 0.025 2.5.0.0 0.020 2.1.1.0 0.010 2.3.1.0 0.010 2.4.0.0 *0.005 1.0.0.0* 0.005 2.2.3.0 0.005 2.0.1.0 *0.005 3.2.0.0* *0.005 4.1.0.0* *0.005 3.3.0.0* *0.005 3.0.0.0* 0.005 2.5.1.0 *0.005 4.4.0.0* 0.005 2.6.0.0 0.005 2.0.2.0 2.2.1.0 (54) 0.519 2.2.1.0 0.315 2.2.0.0 0.074 2.2.2.0 0.037 2.1.0.0 0.019 2.3.1.0 0.019 2.1.2.0 0.019 2.4.0.0 3.0.0.0 (385) 0.844 3.0.0.0 0.075 3.0.0.1 *0.021 2.0.0.0* *0.013 4.0.0.0* 0.010 3.0.0.2 *0.010 5.0.0.0* *0.008 6.0.0.0* *0.005 1.0.0.0* *0.003 4.0.0.1* *0.003 9.0.0.0* 0.003 3.0.0.3 *0.003 4.0.0.2* 0.003 3.0.0.4 3.0.0.1 (76) 0.526 3.0.0.1 0.434 3.0.0.0 0.039 3.0.0.2 3.1.0.0 (230) 0.865 3.1.0.0 0.061 3.1.0.1 0.022 3.1.0.2 0.017 3.2.0.0 *0.009 4.1.0.0* 0.009 3.3.0.0 *0.004 5.1.0.0* *0.004 7.1.0.1* 0.004 3.4.0.0 0.004 3.4.0.1 3.1.1.0 (91) 0.846 3.1.1.0 0.066 3.1.2.0 0.044 3.1.1.1 0.033 3.1.1.2 0.011 3.1.3.0 3.2.0.0 (106) 0.717 3.2.0.0 0.085 3.3.0.0 0.047 3.1.0.0 0.019 3.5.0.0 *0.019 2.1.0.0* 0.019 3.2.0.1 *0.009 4.2.0.1* *0.009 9.2.0.1* 0.009 3.2.0.2 0.009 3.4.0.0 0.009 3.1.0.4 *0.009 5.2.0.0* *0.009 4.2.0.0* *0.009 2.2.0.1* 0.009 3.2.0.4 *0.009 2.4.0.0* 3.3.0.0 (61) 0.672 3.3.0.0 0.115 3.2.0.0 0.049 3.3.0.1 0.049 3.4.0.0 0.049 3.1.0.0 0.016 3.5.0.0 0.016 3.1.0.1 0.016 3.4.0.1 0.016 3.1.0.3 4.0.0.0 (173) 0.734 4.0.0.0 0.046 4.0.0.1 *0.029 2.0.0.0* 0.029 4.1.0.0 *0.029 3.0.0.0* *0.017 7.0.0.0* *0.012 5.0.1.0* 0.012 4.0.2.0 *0.012 5.0.0.0* 0.006 4.0.0.4 0.006 4.0.3.0 0.006 4.0.1.0 0.006 4.0.2.1 0.006 4.0.1.2 0.006 4.0.0.2 *0.006 6.0.0.0* 0.006 4.1.0.1 *0.006 3.1.0.0* *0.006 5.0.0.1* *0.006 3.0.0.1* 0.006 4.4.0.0 *0.006 5.2.0.0* *0.006 2.1.3.0* 4.1.0.0 (121) 0.802 4.1.0.0 0.050 4.0.0.0 0.050 4.1.0.1 0.017 4.2.0.0 0.017 4.1.1.0 *0.008 5.1.0.0* 0.008 4.0.1.0 0.008 4.1.0.2 0.008 4.1.0.3 *0.008 6.1.0.0* 0.008 4.4.0.0 0.008 4.1.1.3 0.008 4.3.1.0 5.0.0.0 (76) 0.724 5.0.0.0 *0.066 3.0.0.0* *0.053 4.0.0.0* 0.039 5.0.1.0 *0.026 2.0.0.0* *0.026 6.0.0.0* *0.013 14.0.0.0* *0.013 9.0.0.0* *0.013 1.0.0.0* *0.013 7.0.0.0* *0.013 3.0.1.0* 6.0.0.0 (109) 0.734 6.0.0.0 0.046 6.0.0.1 0.037 6.0.1.0 *0.028 3.0.0.0* *0.018 4.0.0.0* *0.018 2.0.0.0* *0.018 5.0.0.0* 0.009 6.0.2.0 0.009 6.0.1.1 0.009 6.1.0.1 *0.009 7.0.0.0* 0.009 6.0.0.3 *0.009 5.0.0.1* *0.009 8.0.0.1* 0.009 6.0.3.1 0.009 6.2.0.0 *0.009 10.1.0.0* 0.009 6.1.0.0
 2012-03-15, 23:38 #7 fivemack (loop (#_fork))     Feb 2006 Cambridge, England 22·32·179 Posts And here are the figures for 151-to-160-digit numbers Code: 1.1.0.0 (71) 0.859 1.1.0.0 0.056 1.1.1.0 0.028 1.3.0.0 0.028 1.1.0.1 0.014 1.2.0.0 0.014 1.1.3.0 1.2.0.0 (56) 0.786 1.2.0.0 0.071 1.2.0.1 0.054 1.3.0.0 0.036 1.1.0.0 0.018 1.2.1.0 0.018 1.4.0.0 0.018 1.2.1.1 2.0.0.1 (53) 0.792 2.0.0.1 0.075 2.0.1.1 0.038 2.1.0.1 0.038 2.0.0.2 0.019 2.0.3.1 0.019 2.0.2.1 0.019 2.2.0.1 2.1.0.1 (76) 0.789 2.1.0.1 0.053 2.1.0.2 0.053 2.0.0.1 0.026 2.2.0.1 0.026 2.1.1.1 0.013 2.1.2.1 0.013 2.4.0.1 0.013 2.3.0.1 0.013 2.1.1.2 (these are only the prefixes that occur 50 or more times) Last fiddled with by fivemack on 2012-03-15 at 23:39
2012-03-16, 00:15   #8
science_man_88

"Forget I exist"
Jul 2009
Dumbassville

26×131 Posts

Quote:
 Originally Posted by fivemack A.B.C.D means 2^A*3^B*5^C*7^D These are from 1250 lines from the 2^2*3 project Code: 2.0.0.0 0.890 2.0.0.0 0.044 2.0.1.0 *0.015 3.0.0.0* *0.015 1.0.0.0* 0.011 2.0.2.0 0.007 2.1.0.0 *0.004 7.0.1.0* 0.004 2.0.7.0 *0.004 4.0.0.0* *0.004 3.2.0.0* 0.004 2.0.3.0 2.1.0.0 0.832 2.1.0.0 0.052 2.0.0.0 0.047 2.1.1.0 0.044 2.2.0.0 0.010 2.3.0.0 0.008 2.1.2.0 0.003 2.0.1.0 0.003 2.2.1.0 0.003 2.4.0.0 2.1.1.0 0.608 2.1.1.0 0.278 2.1.0.0 0.089 2.1.2.0 0.013 2.1.3.0 0.013 2.2.1.0 2.2.0.0 0.620 2.2.0.0 0.092 2.1.0.0 0.063 2.2.1.0 0.049 2.3.0.0 0.042 2.0.0.0 0.021 2.2.2.0 0.021 2.1.1.0 0.021 2.5.0.0 0.014 2.2.3.0 *0.007 1.0.0.0* 0.007 2.0.1.0 0.007 2.3.1.0 0.007 2.4.0.0 *0.007 3.0.0.0* 0.007 2.5.1.0 *0.007 4.4.0.0* 0.007 2.6.0.0 Read this as 'if you have currently a factor 2^2*3^2 (and are about 115 digits), probability is about 0.092 that you will be 2^2*3 next iteration'. The probabilities are taken from at least 50 (about 300 for 2.1.0.0) lines. Ones with stars are where the power of two changed.
one thing that came up for me when you sorted it vertically is that:

1.0.0.0 the probabilities for this add up to more than 1 from the addition I did by hand and in a check with PARI.

 2012-03-16, 00:36 #9 Batalov     "Serge" Mar 2008 Phi(4,2^7658614+1)/2 3·5·641 Posts I just recently saw a 2^2*3^2*P -> 2 transition in the wild (967170:i729) and was quite surprised. After a small sampling simulation, I saw that it was indeed quite rare, and your data shows the same (for sure): 2.2.0.0 -> *0.007 1.0.0.0* I concur.
2012-03-16, 08:49   #10
fivemack
(loop (#_fork))

Feb 2006
Cambridge, England

22×32×179 Posts

Quote:
 1.0.0.0 the probabilities for this add up to more than 1
Yes, they do; they're all rounded independently and there are eleven of them. If you want to convert them to the integer numbers of 345ths that they represent and add them up, they will get to 345/345.

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