[seqfan] request for help with a sequence

[John Erickson]

Hi All,

Just joined seqfan list 30 years too late. I never noticed it before and in
particular I never noticed that you can request assistance.
I have recently considered the sequence a(n) defined as

The number of distinct transitive subgroups of S_n, counting conjugates as
distinct.

With the first 13 terms starting at n = 1 given by
1, 1, 2, 9, 20, 279, 512, 19087, 71602, 636365, 1517042, 321965982,
240609602


transitiveSubgroupsCount := function(n)
    local numTransitiveSubgroups, G, cc, class;
    numTransitiveSubgroups := 0;
    G := SymmetricGroup(n);
    cc:=ConjugacyClassesSubgroups(G);
    for class in cc do
        if  IsTransitive(Representative(class), [1..n]) then
               numTransitiveSubgroups := numTransitiveSubgroups +
Size(class);
        fi;
    od;
return numTransitiveSubgroups;
end;

I have proven that A005432(p)-a(p) == 1 (mod p) if p is prime and
based on n<= 13,  I have conjectured that floor(log(A005432(n)/a(n))) <=
(n-1)/2 for n>=1, with equality holding for n>=3 and prime.

Obviously, with only 13 examples this may be ridiculous so I would like
more data. Can anyone suggest better GAP code that uses less memory? It's
my first GAP program as Mathematica was not well suited for this task.

In general I would appreciate any suggestions. If I am doing something
particularly inept, feel free to let me know. Thanks.

John Erickson