%I #21 Sep 15 2024 20:22:57
%S 1,2,6,28,260
%N Integers k such that the maximum number of subgroups of a group of order k is exactly k.
%C Intersection of A368538 and A370422. Difference of A370422 and A370421.
%C a(6) > 2000 if it exists.
%o (Magma) // to get the terms up to 1023.
%o i:=1;
%o while i lt 1024 do // terms up to 1023
%o allGroupsHaveLessThanOrEqualNumberOfSubgroups:=1;
%o someGroupWithExactNumberOfSubgroups:=0;
%o j:=1;
%o while j le NumberOfSmallGroups(i) do //iterate through all the groups of order i
%o G:=SmallGroup(i, j);
%o if #AllSubgroups(G) eq i then
%o someGroupWithExactNumberOfSubgroups:=1;
%o end if;
%o if #AllSubgroups(G) gt i then //some group has > i subgroups
%o allGroupsHaveLessThanOrEqualNumberOfSubgroups:=0;
%o break;
%o end if;
%o j:=j+1;
%o end while;
%o if allGroupsHaveLessThanOrEqualNumberOfSubgroups eq 1 and someGroupWithExactNumberOfSubgroups eq 1 then
%o i;
%o end if;
%o i:=i+1;
%o end while;
%Y Cf. A368538, A370421, A370422.
%K nonn,more
%O 1,2
%A _Robin Jones_, Feb 18 2024