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

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

Offset: 1

Author

Richard Stanley, Jun 10 2025

Keywords

• nonn
• more

Comments

Let 1=b_1A368538). Then a(n) is the number of such groups (up to isomorphism) of order b_n.

Examples

			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.
		

Links

• Dave Benson, Congruence mod four of the number of subgroups of a finite 2-group, discussion in MathOverflow, 2025 Jun 11.

Crossrefs

Cf. A368538, A384727.

Programs

Magma

// Output of A368538(n) and a(n)
limit := 104;
for i in [1 .. limit] do
  j := 0;
  for G in SmallGroups(i) do
    if #AllSubgroups(G) eq i then j +:= 1; end if;
  end for;
  if j gt 0 then i, j; end if;
end for; // Hugo Pfoertner, Jun 10 2025

Maple

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]:
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

Extensions

a(25)-a(32) from Richard Stanley, Jun 11 2025 using results by Dave Benson in MathOverflow discussion.