[seqfan] Vertices, edges, primes and nonprimes
Eric Angelini
Eric.Angelini at kntv.be
Wed Sep 16 15:11:42 CEST 2015
Hello SeqFans,
Let's build a "graph" step by step.
General rules:
(1) an edge links two integers "a" and "b" if
("a" is prime and "b" is nonprime) or if
("a" is nonprime and "b" is prime);
EXAMPLE: 12 or 74 are ok. 1337 is not.
(2) from vertex "n" always leave "n" edges;
w...

EXAMPLE: 1243y...
/ \
x... z...
"w" and "x" are primes, "y" and "z" are
nonprimes.
We see above that from vertex "1" leaves only
1 link; from vertex "2" leave 2 links, from
vertex "3" leave 3 links, from vertex "4" leave
4 links, etc.
(3) always try to link a vertex "n" with a vertex < "n"
(this is the "maximum compactness" condition)

If we start to draw the graph step by step, we'll get:
12
124
3
/
1245
\
7
36
/ \
12458
\
7
etc.
We see above that "5" is linked to 4 and 8; it will be
linked to "6" later and to 9 and 10 which are not yet
present. In the end, "5" will be linked 5 times to
nonprime integers that build the set [4,6,8,9,10]
The first line of the hereunder array is "n".
The vertical line under "n" is the list of the vertices
it is linked to:
n = 1 2 3 4 5 6 7 8 9 10 11 ...
2 1 4 2 4 3 4 3 5 5 6 ...
4 6 3 6 5 6 5 7 7 8 ...
8 5 8 7 8 7 11 11 9 ...
7 9 11 9 11 13 13 10 ...
10 13 10 13 17 17 12 ...
17 12 17 19 19 14 ...
14 19 23 23 15 ...
23 29 29 16 ...
31 31 18 ...
37 20 ...
21 ...
The descending diagonal is not in the OEIS (2,4,8,7,10,...)
Best,
É.
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