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-   -   Which are the "worst" drivers? (https://www.mersenneforum.org/showthread.php?t=16932)

Dubslow 2012-06-24 15:06

Which are the "worst" drivers?
 
So I'm sure we've all read Clifford's [URL="http://dubslow.tk/aliquot/analysis.html"]analysis[/URL] page.

(Btw, I've created a mirror of that page and his "[URL="http://dubslow.tk/aliquot/aliquot.html"]From the Trenches[/URL]" page so that we don't have to deal with the archive, and so that someone could edit it should they feel so inclined. Of course, if Clifford makes his own comeback, I'll take these down.)

Anyways, the analysis classifies the drivers by their characteristics, by how "stable" they are.

Of course, that doesn't take into account other factors, like how much a driver increases a sequence, or how rare getting a 127^2 is. So, since I am not an expert (or even half-competent) in these matters, I'm asking: Is it possible to create an ordered list of drivers from "worst to first"? Obviously the downdriver would be "first". I imagine 2^2 would be next, followed by 2^3, since they tend to drop a sequence, albeit slowly (and they're not very stable).
[code]Best
2
2^2
2^3
...
?
...
2^6 * 127 ??
Worst[/code]
(PS Are there any drivers which are missing from that table?)

Batalov 2012-06-24 19:44

Which are the "worst" drivers?
 
Reminds me of a trick question:
[QUOTE]Q: How can you spot an intoxicated Asian Female driver?

[SPOILER]A: You can't![/SPOILER][/QUOTE]

science_man_88 2012-06-24 23:34

[QUOTE=Dubslow;303173]So I'm sure we've all read Clifford's [URL="http://dubslow.tk/aliquot/analysis.html"]analysis[/URL] page.

(Btw, I've created a mirror of that page and his "[URL="http://dubslow.tk/aliquot/aliquot.html"]From the Trenches[/URL]" page so that we don't have to deal with the archive, and so that someone could edit it should they feel so inclined. Of course, if Clifford makes his own comeback, I'll take these down.)

Anyways, the analysis classifies the drivers by their characteristics, by how "stable" they are.

Of course, that doesn't take into account other factors, like how much a driver increases a sequence, or how rare getting a 127^2 is. So, since I am not an expert (or even half-competent) in these matters, I'm asking: Is it possible to create an ordered list of drivers from "worst to first"? Obviously the downdriver would be "first". I imagine 2^2 would be next, followed by 2^3, since they tend to drop a sequence, albeit slowly (and they're not very stable).
[code]Best
2
2^2
2^3
...
?
...
2^6 * 127 ??
Worst[/code]
(PS Are there any drivers which are missing from that table?)[/QUOTE]

[CODE]for(a=1,60,for(b=1,#divisors(sigma(2^a)),if(sigma(divisors(sigma(2^a))[b])%(2^(a-1))==0,print(2"^"a"*"divisors(sigma(2^a))[b]))))[/CODE] according to this code:

[QUOTE]2^1*1
2^1*3
2^2*7
2^3*3
2^3*15
2^4*31
2^5*21
2^6*127
2^9*1023
2^12*8191
2^16*131071
2^18*524287
2^30*2147483647
2^60*2305843009213693951[/QUOTE]

are all drivers but not all of the possible drivers, partly because I can't search higher how I have things set up. the bigger a driver is the rarer it likely is. as to biggest gains I would think the larger ones but all I know is possibly with numbers that are driver * prime.

fivemack 2012-06-24 23:55

Here's an analysis over the 115-to-125-digit elements of all the sequences I've run:

a.b.c.d.e means 2^a * 3^b * 5^c * 7^d * 31^e

Numbers in brackets at the beginning are the number of lines I saw with that exponent combination.

The first probability is the chance of going from one exponent-set to another, so a measure of stability.

Numbers in brackets at the end are the average change in log(K) of such lines.

So: powers of three are the annoying ones, the worst driver for making sequences get larger is 2^2*3^2*5, 2^3*3 is beautifully stable and drives terms up fast. Powers of two with no other small primes get less stable as the exponent increases.

[code]

1.0.0.0.0 (564) 0.901 1.0.0.0.0 (-0.245)
0.053 1.0.1.0.0 (-0.262)
0.023 1.0.0.0.1 (-0.242)

1.0.1.0.0 (99) 0.586 1.0.1.0.0 (-0.054)
0.263 1.0.0.0.0 (-0.044)
0.111 1.0.2.0.0 (-0.016)

1.1.0.0.0 (433) 0.857 1.1.0.0.0 (0.045)
0.046 1.1.1.0.0 (0.041)
0.032 1.2.0.0.0 (0.018)

1.1.1.0.0 (69) 0.507 1.1.1.0.0 (0.180)
0.304 1.1.0.0.0 (0.171)

1.2.0.0.0 (209) 0.756 1.2.0.0.0 (0.108)
0.057 1.1.0.0.0 (0.094)
0.057 1.3.0.0.0 (0.101)

1.2.1.0.0 (60) 0.700 1.2.1.0.0 (0.226)
0.217 1.2.0.0.0 (0.232)

1.3.0.0.0 (91) 0.692 1.3.0.0.0 (0.132)
0.132 1.2.0.0.0 (0.108)

2.0.0.0.0 (2003) 0.846 2.0.0.0.0 (-0.078)
0.057 2.0.1.0.0 (-0.082)
0.024 2.0.0.0.1 (-0.079)

2.0.0.0.1 (67) 0.776 2.0.0.0.0 (-0.049)

2.0.1.0.0 (414) 0.580 2.0.1.0.0 (0.085)
0.280 2.0.0.0.0 (0.078)
0.075 2.0.2.0.0 (0.075)
0.031 2.0.1.0.1 (0.076)

2.0.2.0.0 (84) 0.321 2.0.2.0.0 (0.124)
0.286 2.0.0.0.0 (0.103)
0.274 2.0.1.0.0 (0.110)

2.1.0.0.0 (1130) 0.795 2.1.0.0.0 (0.156)
0.052 2.0.0.0.0 (0.143)
0.045 2.2.0.0.0 (0.140)
0.039 2.1.1.0.0 (0.149)
0.022 2.1.0.0.1 (0.161)
0.019 2.3.0.0.0 (0.149)

2.1.1.0.0 (209) 0.574 2.1.1.0.0 (0.277)
0.278 2.1.0.0.0 (0.273)
0.096 2.1.2.0.0 (0.273)
0.014 2.1.3.0.0
0.010 2.1.0.0.1
0.010 2.2.1.0.0
0.010 2.1.1.0.1
0.005 2.1.1.0.2
0.005 2.2.0.0.0

2.1.2.0.0 (53) 0.321 2.1.2.0.0 (0.300)
0.245 2.1.0.0.0 (0.312)
0.226 2.1.1.0.0 (0.294)

2.2.0.0.0 (432) 0.650 2.2.0.0.0 (0.218)
0.086 2.1.0.0.0 (0.203)
0.049 2.0.0.0.0 (0.195)
0.049 2.2.1.0.0 (0.216)
0.046 2.3.0.0.0 (0.196)

2.2.1.0.0 (106) 0.500 2.2.1.0.0 (0.339)
0.274 2.2.0.0.0 (0.338)

2.3.0.0.0 (130) 0.538 2.3.0.0.0 (0.231)
0.154 2.2.0.0.0 (0.220)
0.123 2.1.0.0.0 (0.211)

3.0.0.0.0 (975) 0.907 3.0.0.0.0 (-0.014)
0.029 3.0.0.0.1 (-0.013)
*0.021 2.0.0.0.0* (-0.052)
*0.019 4.0.0.0.0* (-0.049)

3.0.1.0.0 (265) 0.879 3.0.1.0.0 (0.130)
0.049 3.0.2.0.0 (0.133)

3.0.2.0.0 (58) 0.621 3.0.2.0.0 (0.155)
0.293 3.0.1.0.0 (0.154)

3.1.0.0.0 (572) 0.946 3.1.0.0.0 (0.217)
0.024 3.1.0.0.1 (0.209)

3.1.1.0.0 (178) 0.888 3.1.1.0.0 (0.331)
0.073 3.1.2.0.0 (0.315)

3.2.0.0.0 (179) 0.771 3.2.0.0.0 (0.263)
0.067 3.1.0.0.0 (0.274)

3.3.0.0.0 (75) 0.587 3.3.0.0.0 (0.291)
0.160 3.2.0.0.0 (0.258)

4.0.0.0.0 (443) 0.767 4.0.0.0.0 (0.018)
0.077 4.0.1.0.0 (0.016)
*0.027 3.0.0.0.0* (-0.019)

4.0.0.0.1 (129) 0.860 4.0.0.0.1 (0.051)

4.0.1.0.0 (132) 0.583 4.0.1.0.0 (0.160)
0.265 4.0.0.0.0 (0.142)

4.1.0.0.0 (216) 0.782 4.1.0.0.0 (0.239)
0.060 4.0.0.0.0 (0.208)
0.051 4.2.0.0.0 (0.232)

4.1.0.0.1 (109) 0.798 4.1.0.0.1 (0.266)

4.2.0.0.0 (64) 0.625 4.2.0.0.0 (0.299)
0.172 4.1.0.0.0 (0.281)

4.2.0.0.1 (53) 0.604 4.2.0.0.1 (0.303)

5.0.0.0.0 (247) 0.745 5.0.0.0.0 (0.054)

5.1.0.0.0 (151) 0.828 5.1.0.0.0 (0.280)
0.093 5.1.1.0.0 (0.264)

6.0.0.0.0 (179) 0.721 6.0.0.0.0 (0.036)

6.1.0.0.0 (60) 0.750 6.1.0.0.0 (0.248)
[/code]

Dubslow 2012-06-25 00:12

Wow, that's an amazing analysis. What original form do you have it in, i.e. would it be easy for you to make versions sorted by stability and/or rate of change, or perhaps just send it to me if it's useful? Considering the amount of effort you must have already put into such a table, I'd be more than happy to effect the sorts myself, but I figure I should ask and see what you've got.

Thanks! :smile:

Edit: The more I think about this, the more I think this information is very useful/enlightening, and deserves its own webpage and formatting. What do you think?

fivemack 2012-06-25 06:20

This is, you will be unsurprised to hear, the output from a perl script run over /home/nfsworld on my large machine (17 megabytes, 87klines); it could relatively easily be rearranged.

I'm not quite sure to what extent the probabilities vary with number size (obviously, since the files I'm using to test include the output from the work in [url]http://www.mersenneforum.org/showpost.php?p=301237&postcount=252[/url], I have rather more lines for numbers of size 115-125 digits).

I would suggest working out what sort of HTML output you want, making a couple of lines of example, and I'll change the perl (the alternative is that I put a large .tgz file of aliquot-sequence-output and the perl up on [url]www.fivemack.org[/url] somewhere and you can do the stats yourself). But I am very busy this week.

Dubslow 2012-06-25 20:11

The second option sounds easier for you, I'll be happy to do it myself. When it comes to HTML, being that I have far less experience with that than Python or C (and I have very little experience with the last two compared to most others here), I tend to fiddle a [i]lot[/i] with HTML, even after I would have sent you the 'final' format, so it's probably best that I modify the Perl myself. :smile:

Dubslow 2012-07-21 23:18

Look what I found :smile:

[url]http://www.mersenneforum.org/showthread.php?t=12597[/url]
:hello: Greebley!



(fivemack...?)

schickel 2012-07-22 05:39

1 Attachment(s)
[QUOTE=Dubslow;305451]Look what I found :smile:

[url]http://www.mersenneforum.org/showthread.php?t=12597[/url]
:hello: Greebley!



(fivemack...?)[/QUOTE]Heh....forgot about that one.

I think drivers could actually be considered bad or persistent, or both. Bad would be in terms of increasing the size of a sequences; persistent would be in terms of chances of escape.

For example, how would you characterize this upward run (starting at ~i5154)?

[SPOILER]Hint: It's 2^6 * 127 all the way.....[/SPOILER]


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