# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a368538 Showing 1-1 of 1 %I A368538 #47 Feb 18 2024 11:47:03 %S A368538 1,2,6,8,28,40,48,54,72,96,100,104,128,132,144,160,176,180,192,216, %T A368538 240,252,260,288,324,336,368,384,416,456,480,496 %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(33) >= 512. The smallest term strictly larger than 512 is 560. -_Robin Jones_, Feb 18 2024 %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. %K A368538 nonn,more %O A368538 1,2 %A A368538 _Robin Jones_, Dec 29 2023 # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE