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Old 2011-04-21, 10:21   #1
Wini
 
Apr 2011
Baden, Germany

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Question 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 ...)
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Old 2011-04-21, 12:23   #2
kar_bon
 
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Wolfgang Creyaufmueller is a good source to find and get some deeper information in Aliquot sequences.
See his page here and a stats-page for families can be found here.
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Old 2011-04-21, 12:26   #3
Raman
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Of course, for the complete list of that - you must go to
http://www.aliquot.de/archiv/1m.zip

Last fiddled with by Raman on 2011-04-21 at 12:30
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Old 2011-04-21, 17:49   #4
Raman
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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

Last fiddled with by Raman on 2011-04-21 at 17:59
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Old 2011-04-21, 22:15   #5
Batalov
 
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Quote:
Originally Posted by Wini View Post
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 ...)
Most tables don't bother with odd number sequences, but
you can generate your own simple table by
pari/gp> for (n=2,10000000,write("mytable1",n," ",sigma(n)-n))
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)...
This is similar to EndGame TableBases.

Incidentally, the OEIS sequence A048138 will describe the number of entries into a node n (offset of this sequence is 2).

For visualization, you may want to try something like Cytoscape.

Last fiddled with by Batalov on 2011-04-21 at 22:54 Reason: some nodes/curios
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Old 2011-04-22, 20:05   #6
Wini
 
Apr 2011
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Quote:
Originally Posted by Batalov View Post
Most tables don't bother with odd number sequences, but
you can generate your own simple table by
pari/gp> for (n=2,10000000,write("mytable1",n," ",sigma(n)-n))
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)...
This is similar to EndGame TableBases.

Incidentally, the OEIS sequence A048138 will describe the number of entries into a node n (offset of this sequence is 2).

For visualization, you may want to try something like Cytoscape.
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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