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

1, 2, 6, 8, 28, 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

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..32.

Example

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.

Prog

(Magma, to get the terms up to 100)
i:=1;
while i lt 100 do  // terms up to 100
for G in SmallGroups(i) do
if #AllSubgroups(G) eq i then
    i; break;
end if;
; end for;
i:=i+1;
end while;

Crossrefs

Cf. A018216, A061034.

Sequence in context: A180814 A290679 A290423 * A272614 A210737 A140539

Adjacent sequences: A368535 A368536 A368537 * A368539 A368540 A368541

Keyword

nonn,more

Author

Robin Jones, Dec 29 2023

Status

approved