A370422 -id:A370422 - cp's OEIS frontend

A370421 Integers k such that all groups of order k have strictly fewer than k subgroups.

3, 5, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 29, 30, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 85, 86, 87, 89, 91, 92, 93, 94, 95, 97, 99

Offset: 1

Robin Jones, Feb 18 2024

nonn

This sequence is infinite. All primes other than 2 appear in the sequence.

Robin Jones, Table of n, a(n) for n = 1..782

Cf. A368538, A370422.

// to get the terms up to 1023. The program will not work for i=1024, returning a positive result, since those groups are not classified.
i:=1;
while i lt 1024 do  // terms up to 1023
    inSequence:=1;
    j:=1;
    while j le NumberOfSmallGroups(i) do //iterate through all the groups of order i
        G:=SmallGroup(i,j);
        if #AllSubgroups(G) ge i then //some group has >= i subgroups
            inSequence:=0;
            break;
        end if;
        j:=j+1;
    end while;
    if inSequence eq 1 then
        i;
    end if;
    i:=i+1;
end while;

A370423 Integers k such that the maximum number of subgroups of a group of order k is exactly k.

Original entry on oeis.org

1, 2, 6, 28, 260

Offset: 1

Robin Jones, Feb 18 2024

Intersection of A368538 and A370422. Difference of A370422 and A370421.

a(6) > 2000 if it exists.

Cf. A368538, A370421, A370422.

// to get the terms up to 1023.
i:=1;
while i lt 1024 do  // terms up to 1023
allGroupsHaveLessThanOrEqualNumberOfSubgroups:=1;
someGroupWithExactNumberOfSubgroups:=0;
j:=1;
while j le NumberOfSmallGroups(i) do //iterate through all the groups of order i
G:=SmallGroup(i, j);
if #AllSubgroups(G) eq i then
    someGroupWithExactNumberOfSubgroups:=1;
end if;
if #AllSubgroups(G) gt i then //some group has > i subgroups
    allGroupsHaveLessThanOrEqualNumberOfSubgroups:=0;
    break;
end if;
j:=j+1;
end while;
if allGroupsHaveLessThanOrEqualNumberOfSubgroups eq 1 and someGroupWithExactNumberOfSubgroups eq 1 then
    i;
end if;
i:=i+1;
end while;

cp's OEIS Frontend

A370421 Integers k such that all groups of order k have strictly fewer than k subgroups.

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