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-   -   visualizing prime families (https://www.mersenneforum.org/showthread.php?t=15538)

Wini 2011-04-21 10:21

visualizing prime families
 
Hi
I would like to visualize prime families (the incoming ends of aliquot sequences) to some of the more popular primes like f.e. 43.
Actually there are 2 points where I need some help. If not in the mersenneforum, where else could I hope to find someone to be able ;-))
cheers Wini

1. Does anyone know existing graphs visualizing prime families?
(right now I use MS Powerpoint)

2. Does anyone dispose of a matrix of terminated aliquot sequences with the attributes #start number, #target (prime, amicable number, perfect number), #index (how many elements left in the aliquot sequence to the target), #very next element in the aliquot sequence
(right now it`s a lot of work to find out :blush:...)

kar_bon 2011-04-21 12:23

Wolfgang Creyaufmueller is a good source to find and get some deeper information in Aliquot sequences.
See his page [url=http://www.aliquot.de/]here[/url] and a stats-page for families can be found [url=http://www.aliquot.de/tabellen/1-1m.txt]here[/url].

Raman 2011-04-21 12:26

Of course, for the complete list of that - you must go to
[URL]http://www.aliquot.de/archiv/1m.zip[/URL]

Raman 2011-04-21 17:49

Aliquot sequence
30 terminates in 3
60 terminates in 43
96 terminates in 37
48 terminates in 41
138 terminates in 59
180 terminates in 601 (210)
11025 terminates in 7
18 terminates in 11
120 terminates in 12161
720 terminates in 277 (2880)
702 terminates in 191
936 terminates in 43
726 terminates in 59 (858)
650 terminates in 496
1235 terminates in 6

These are rather that twelve open end sequences with that starting term that is below 1000 in any case
276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996

Batalov 2011-04-21 22:15

[QUOTE=Wini;259184]Hi
I would like to visualize prime families (the incoming ends of aliquot sequences) to some of the more popular primes like f.e. 43.
Actually there are 2 points where I need some help. If not in the mersenneforum, where else could I hope to find someone to be able ;-))
cheers Wini

1. Does anyone know existing graphs visualizing prime families?
(right now I use MS Powerpoint)

2. Does anyone dispose of a matrix of terminated aliquot sequences with the attributes #start number, #target (prime, amicable number, perfect number), #index (how many elements left in the aliquot sequence to the target), #very next element in the aliquot sequence
(right now it`s a lot of work to find out :blush:...)[/QUOTE]
Most tables don't bother with odd number sequences, but
you can generate your own simple table by
[FONT=System]pari/gp> for (n=2,10000000,write("mytable1",n," ",sigma(n)-n))[/FONT]
and then analyze the output for cycles, multiple entries into a node etc...

After the sequence which ends with 6 was found, I played with this idea for a while. There are some bottlenecks in some descents, and some others are highly branched. Note: 28 is unreacheable (except from itself); it is a singleton. 6 and 496 have an entry and a cycle into itself... etc etc etc. Some highly branched nodes are
[code]129 2731
127 2941
117 2971
116 2521
115 2311
115 2851
113 2881
111 2641
109 2761

...(there are undoubtedly nodes with more entries)...[/code]
This is similar to EndGame TableBases.

Incidentally, the OEIS sequence [URL="http://oeis.org/A048138"]A048138[/URL] will describe the number of entries into a node [I]n[/I] (offset of this sequence is 2).

For visualization, you may want to try something like [URL="http://cytoscape.wodaklab.org/wiki/"]Cytoscape[/URL].

Wini 2011-04-22 20:05

[QUOTE=Batalov;259237]Most tables don't bother with odd number sequences, but
you can generate your own simple table by
[FONT=System]pari/gp> for (n=2,10000000,write("mytable1",n," ",sigma(n)-n))[/FONT]
and then analyze the output for cycles, multiple entries into a node etc...

After the sequence which ends with 6 was found, I played with this idea for a while. There are some bottlenecks in some descents, and some others are highly branched. Note: 28 is unreacheable (except from itself); it is a singleton. 6 and 496 have an entry and a cycle into itself... etc etc etc. Some highly branched nodes are
[code]129 2731
127 2941
117 2971
116 2521
115 2311
115 2851
113 2881
111 2641
109 2761

...(there are undoubtedly nodes with more entries)...[/code]
This is similar to EndGame TableBases.

Incidentally, the OEIS sequence [URL="http://oeis.org/A048138"]A048138[/URL] will describe the number of entries into a node [I]n[/I] (offset of this sequence is 2).

For visualization, you may want to try something like [URL="http://cytoscape.wodaklab.org/wiki/"]Cytoscape[/URL].[/QUOTE]
Hey,
thanks indeed. There is such a lot of input.
The Aliquot-pages to find the targets, the Pari/GP-function to find the sigma-lists, the OEIS-reference to check the nodes (wow, I never thought of nodes with such a lot of entries - hmm). It makes it much easier to finish the data for my prime family project.
I`ll try Cytoscape to visualize the results and will be back with some graphs - in due course ;-)
take care, cheers, Wini


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