cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A384800 a(n) = A384727(A368538(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 6, 1, 1, 7, 1, 1, 5, 1, 6, 1, 1, 2, 4, 1, 1, 23, 1, 13, 2
Offset: 1

Views

Author

Richard Stanley, Jun 10 2025

Keywords

Comments

Let 1=b_1A368538). Then a(n) is the number of such groups (up to isomorphism) of order b_n.

Examples

			Of the groups of order at most six, the 1-element group, 2-element group, and the symmetric group S_3 of order six are the only ones with the same number of elements as subgroups. Hence a(1) = a(2) = a(3) = 1.

Links

Crossrefs

Cf. A368538, A384727.

Programs

  • Magma
    // Output of A368538(n) and a(n)
limit := 104;
for i in [1 .. limit] do
  j := 0;
  for G in SmallGroups(i) do
    if #AllSubgroups(G) eq i then j +:= 1; end if;
  end for;
  if j gt 0 then i, j; end if;
end for; // Hugo Pfoertner, Jun 10 2025
  • Maple
    A368538:= [1, 2, 6, 8, 28, 36, 40, 48, 54, 72, 96, 100, 104, 128, 132, 144, 160, 176, 180, 192, 216, 240, 252, 260, 288, 324, 336, 368, 384, 416, 456, 480, 496]:
seq(nops(select(g -> nops(convert(SubgroupLattice(g),list))=k, [seq(SmallGroup(k,i),i=1..NumGroups(k))])),k=A368538); # Robert Israel, Jun 10 2025

Extensions

a(25)-a(32) from Richard Stanley, Jun 11 2025 using results by Dave Benson in MathOverflow discussion.

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

Original entry on oeis.org

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

Views

Author

Robin Jones, Feb 18 2024

Keywords

Comments

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

Links

Crossrefs

Cf. A368538, A370422.

Programs

  • Magma
    // 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;

A370422 Integers k such that all groups of order k have at most k subgroups.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 21, 22, 23, 25, 26, 28, 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

Views

Author

Robin Jones, Feb 18 2024

Keywords

Comments

This sequence is infinite. All primes appear in the sequence.

Links

Crossrefs

Cf. A368538, A370421.

Programs

  • Magma
    // 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) gt 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;

A384727 Number of groups of order n (up to isomorphism) with exactly n subgroups.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1
Offset: 1

Views

Author

Richard Stanley, Jun 08 2025

Keywords

Comments

See A384800 for more information.

Examples

			The symmetric group S_3 has six elements and six subgroups. The other group of order six has four subgroups, so a(6)=1.

Links

Crossrefs

Cf. A368538, A384800.

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

Views

Author

Robin Jones, Feb 18 2024

Keywords

Comments

Intersection of A368538 and A370422. Difference of A370422 and A370421.
a(6) > 2000 if it exists.

Crossrefs

Cf. A368538, A370421, A370422.

Programs

  • Magma
    // 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;
Showing 1-5 of 5 results.

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