cp's OEIS Frontend

A384800 a(n) = A384727(A368538(n)).

%I A384800 #21 Jun 14 2025 12:11:12
%S A384800 1,1,1,1,1,1,2,1,1,1,1,1,1,6,1,1,7,1,1,5,1,6,1,1,2,4,1,1,23,1,13,2
%N A384800 a(n) = A384727(A368538(n)).
%C A384800 Let 1=b_1<b_2<... be those positive integers k for which there exists a group of order k with exactly k subgroups (A368538). Then a(n) is the number of such groups (up to isomorphism) of order b_n.
%H A384800 Dave Benson, <a href="https://mathoverflow.net/questions/496010/">Congruence mod four of the number of subgroups of a finite 2-group</a>, discussion in MathOverflow, 2025 Jun 11.
%e A384800 Of the groups of order at most six, the 1-element group, 2-element group, and the symmetric group S_3 of order six are the only ones with the same number of elements as subgroups. Hence a(1) = a(2) = a(3) = 1.
%p A384800 A368538:= [1, 2, 6, 8, 28, 36, 40, 48, 54, 72, 96, 100, 104, 128, 132, 144, 160, 176, 180, 192, 216, 240, 252, 260, 288, 324, 336, 368, 384, 416, 456, 480, 496]:
%p A384800 seq(nops(select(g -> nops(convert(SubgroupLattice(g),list))=k, [seq(SmallGroup(k,i),i=1..NumGroups(k))])),k=A368538); # _Robert Israel_, Jun 10 2025
%o A384800 (Magma) // Output of A368538(n) and a(n)
%o A384800 limit := 104;
%o A384800 for i in [1 .. limit] do
%o A384800   j := 0;
%o A384800   for G in SmallGroups(i) do
%o A384800     if #AllSubgroups(G) eq i then j +:= 1; end if;
%o A384800   end for;
%o A384800   if j gt 0 then i, j; end if;
%o A384800 end for; // _Hugo Pfoertner_, Jun 10 2025
%Y A384800 Cf. A368538, A384727.
%K A384800 nonn,more
%O A384800 1,7
%A A384800 _Richard Stanley_, Jun 10 2025
%E A384800 a(25)-a(32) from _Richard Stanley_, Jun 11 2025 using results by Dave Benson in MathOverflow discussion.