Some transition probabilities

fivemack · 2012-03-15, 14:29

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.

firejuggler · 2012-03-15, 16:08

what about those marked with an asterisk? not enough data to be confident?

fivemack · 2012-03-15, 16:57

Ones with stars are where the power of two changed.

firejuggler · 2012-03-15, 18:11

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?

henryzz · 2012-03-15, 23:28

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.

fivemack · 2012-03-15, 23:35

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

fivemack · 2012-03-15, 23:38

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)

science_man_88 · 2012-03-16, 00:15

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.

Batalov · 2012-03-16, 00:36

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.

fivemack · 2012-03-16, 08:49

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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2012-03-15, 16:08	#2
firejuggler "Vincent" Apr 2010 Over the rainbow 5·11·53 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 2·7·461 Posts	Ones with stars are where the power of two changed.

2012-03-16, 00:36	#9
Batalov "Serge" Mar 2008 Phi(4,2^7658614+1)/2 2³×3×5²×17 Posts	I just recently saw a 2^23^2P -> 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.