A370423 - OEIS

%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