cp's OEIS Frontend

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

%I A368538 #63 Jun 14 2025 11:45:30
%S A368538 1,2,6,8,28,36,40,48,54,72,96,100,104,128,132,144,160,176,180,192,216,
%T A368538 240,252,260,288,324,336,368,384,416,456,480,496,560,576,588,624,640,
%U A368538 672,704,720
%N A368538 Integers k such that there exists a group of order k with exactly k subgroups.
%C A368538 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
%C A368538 a(34) >= 512. The smallest term strictly larger than 512 is 560. - _Robin Jones_, Feb 18 2024
%H A368538 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, Jun 11 2025.
%H A368538 Richard Stanley, <a href="https://mathoverflow.net/questions/495845">What finite groups have as many elements as subgroups?</a> Question in MathOverflow, answered by Dave Benson and others, Jun 07 2025.
%e A368538 1 is a term since the trivial group (order 1) has exactly 1 subgroup.
%e A368538 2 is a term since the cyclic group C_2 has exactly 2 subgroups.
%e A368538 6 is a term since the symmetric group S_3 has exactly 6 subgroups.
%o A368538 (Magma, to get the terms up to 100)
%o A368538 i:=1;
%o A368538 while i lt 100 do  // terms up to 100
%o A368538 for G in SmallGroups(i) do
%o A368538 if #AllSubgroups(G) eq i then
%o A368538     i; break;
%o A368538 end if;
%o A368538 ; end for;
%o A368538 i:=i+1;
%o A368538 end while;
%Y A368538 Cf. A018216, A061034, A384727, A384800.
%K A368538 nonn,more
%O A368538 1,2
%A A368538 _Robin Jones_, Dec 29 2023
%E A368538 Missing term 36 added by _Hugo Pfoertner_, Jun 10 2025, following a suggestion by Dave Benson in the MathOverflow discussion.
%E A368538 a(34)-a(41) from _Richard Stanley_, Jun 11 2025, using results by Dave Benson in MathOverflow discussion of question 496010.