A384727 -id:A384727 - cp's OEIS frontend

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

Richard Stanley, Jun 10 2025

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

			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.

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

Cf. A368538, A384727.

// 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

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

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

A368538 Integers k such that there exists a group of order k with exactly k subgroups.

Original entry on oeis.org

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, 560, 576, 588, 624, 640, 672, 704, 720

Offset: 1

Robin Jones, Dec 29 2023

Powers of 4 cannot appear in this sequence. This is because for a group of order p^n, the number of subgroups of order p^k is congruent to 1 mod p, for 0 <= k <= n. It follows from p=2 and Lagrange's theorem that the number of subgroups of order 2^n for n even is congruent to 1 mod 2, i.e. not equal to 2^n. - Robin Jones, Feb 17 2024

a(34) >= 512. The smallest term strictly larger than 512 is 560. - Robin Jones, Feb 18 2024

			1 is a term since the trivial group (order 1) has exactly 1 subgroup.
2 is a term since the cyclic group C_2 has exactly 2 subgroups.
6 is a term since the symmetric group S_3 has exactly 6 subgroups.

Dave Benson, Congruence mod four of the number of subgroups of a finite 2-group, discussion in MathOverflow, Jun 11 2025.
Richard Stanley, What finite groups have as many elements as subgroups? Question in MathOverflow, answered by Dave Benson and others, Jun 07 2025.

Cf. A018216, A061034, A384727, A384800.

Missing term 36 added by Hugo Pfoertner, Jun 10 2025, following a suggestion by Dave Benson in the MathOverflow discussion.

a(34)-a(41) from Richard Stanley, Jun 11 2025, using results by Dave Benson in MathOverflow discussion of question 496010.

cp's OEIS Frontend

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

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