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 1-6 of 6 results.

A173669 Numbers k which divide number of groups of order <= k (A063756).

Original entry on oeis.org

1, 2, 3, 12, 26, 31, 84, 119, 248, 1107, 1543, 1683
Offset: 1

Views

Author

Jonathan Vos Post, Nov 24 2010

Keywords

Comments

No other terms up to 2047. - Eric M. Schmidt, Feb 10 2013

Examples

			a(1) = 1 because 1 divides (the number of groups of order <= 1) = (number of groups of order 1) = 1.
a(4) = 12 because 12 divides (the number of groups of order <= 12) = 24.
a(5) = 26 because 26 divides (the number of groups of order <= 26) = 78 = 3 * 26.
a(6) = 31 because 31 divides (the number of groups of order <= 31) = 93 = 3 * 31.

Crossrefs

Cf. A000001, A063756.

Programs

  • GAP
    A173669 := function(max) local n, res, i; n := 0; res := []; for i in [1..max] do n := n + NrSmallGroups(i); if n mod i = 0 then Add(res, i); fi; od; return res; end; # Eric M. Schmidt, Feb 10 2013

Formula

{k: k | A063756(k)} == {k: k | SUM[i=1..k] A000001(i)}.

Extensions

More terms from Eric M. Schmidt, Feb 10 2013

A000001 Number of groups of order n.

Original entry on oeis.org

0, 1, 1, 1, 2, 1, 2, 1, 5, 2, 2, 1, 5, 1, 2, 1, 14, 1, 5, 1, 5, 2, 2, 1, 15, 2, 2, 5, 4, 1, 4, 1, 51, 1, 2, 1, 14, 1, 2, 2, 14, 1, 6, 1, 4, 2, 2, 1, 52, 2, 5, 1, 5, 1, 15, 2, 13, 2, 2, 1, 13, 1, 2, 4, 267, 1, 4, 1, 5, 1, 4, 1, 50, 1, 2, 3, 4, 1, 6, 1, 52, 15, 2, 1, 15, 1, 2, 1, 12, 1, 10, 1, 4, 2
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy, Dec 16 2004
Also, number of nonisomorphic primitives (antiderivatives) of the combinatorial species Lin[n-1], or X^{n-1}; see Rajan, Summary item (i). - Nicolae Boicu, Apr 29 2011
In (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), a(n) is called the "group number of n", denoted by gnu(n), and the first occurrence of k is called the "minimal order attaining k", denoted by moa(k) (see A046057). - Daniel Forgues, Feb 15 2017
It is conjectured in (J. H. Conway, Heiko Dietrich and E. A. O'Brien, 2008) that the sequence n -> a(n) -> a(a(n)) = a^2(n) -> a(a(a(n))) = a^3(n) -> ... -> consists ultimately of 1s, where a(n), denoted by gnu(n), is called the "group number of n". - Muniru A Asiru, Nov 19 2017
MacHale (2020) shows that there are infinitely many values of n for which there are more groups than rings of that order (cf. A027623). He gives n = 36355 as an example. It would be nice to have enough values of n to create an OEIS entry for them. - N. J. A. Sloane, Jan 02 2021
I conjecture that a(i) * a(j) <= a(i*j) for all nonnegative integers i and j. - Jorge R. F. F. Lopes, Apr 21 2024

Examples

			Groups of orders 1 through 10 (C_n = cyclic, D_n = dihedral of order n, Q_8 = quaternion, S_n = symmetric):
1: C_1
2: C_2
3: C_3
4: C_4, C_2 X C_2
5: C_5
6: C_6, S_3=D_6
7: C_7
8: C_8, C_4 X C_2, C_2 X C_2 X C_2, D_8, Q_8
9: C_9, C_3 X C_3
10: C_10, D_10

References

  • S. R. Blackburn, P. M. Neumann, and G. Venkataraman, Enumeration of Finite Groups, Cambridge, 2007.
  • L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
  • J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
  • H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
  • CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
  • R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, A Foundation for Computer Science, Addison-Wesley Publ. Co., Reading, MA, 1989, Section 6.6 'Fibonacci Numbers' pp. 281-283.
  • M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
  • D. Joyner, 'Adventures in Group Theory', Johns Hopkins Press. Pp. 169-172 has table of groups of orders < 26.
  • D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
  • M. F. Newman and E. A. O'Brien, A CAYLEY library for the groups of order dividing 128. Group theory (Singapore, 1987), 437-442, de Gruyter, Berlin-New York, 1989.
  • N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.
Cf. A046058, A046059, A023675, A023676.
A003277 gives n for which A000001(n) = 1, A063756 (partial sums).
A046057 gives first occurrence of each k.
A027623 gives the number of rings of order n.

Programs

  • GAP
    A000001 := Concatenation([0], List([1..500], n -> NumberSmallGroups(n))); # Muniru A Asiru, Oct 15 2017
  • Magma
    D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // John Cannon, Dec 23 2006
  • Maple
    GroupTheory:-NumGroups(n); # with(GroupTheory); loads this command - N. J. A. Sloane, Dec 28 2017
  • Mathematica
    FiniteGroupCount[Range[100]] (* Harvey P. Dale, Jan 29 2013 *)
a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* Michael Somos, May 28 2014 *)

Formula

From Mitch Harris, Oct 25 2006: (Start)
For p, q, r primes:
a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
a(p^5) = 61 + 2*p + 2*gcd(p-1,3) + gcd(p-1,4), p >= 5, a(2^5)=51, a(3^5)=67.
a(p^e) ~ p^((2/27)e^3 + O(e^(8/3))).
a(p*q) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
a(p*q^2) is one of the following:
---------------------------------------------------------------------------
| a(p*q^2) | p*q^2 of the form | Sequences (p*q^2) |
---------- ------------------------------------------ ---------------------
| (p+9)/2 | q == 1 (mod p), p odd | A350638 |
| 5 | p=3, q=2 => p*q^2 = 12 |Special case with A_4|
| 5 | p=2, q odd | A143928 |
| 5 | p == 1 (mod q^2) | A350115 |
| 4 | p == 1 (mod q), p > 3, p !== 1 (mod q^2) | A349495 |
| 3 | q == -1 (mod p), p and q odd | A350245 |
| 2 | q !== +-1 (mod p) and p !== 1 (mod q) | A350422 |
---------------------------------------------------------------------------
[Table from Bernard Schott, Jan 18 2022]
a(p*q*r) (p < q < r) is one of the following:
q == 1 (mod p) r == 1 (mod p) r == 1 (mod q) a(p*q*r)
-------------- -------------- -------------- --------
No No No 1
No No Yes 2
No Yes No 2
No Yes Yes 4
Yes No No 2
Yes No Yes 3
Yes Yes No p+2
Yes Yes Yes p+4
[table from Derek Holt].
(End)
a(n) = A000688(n) + A060689(n). - R. J. Mathar, Mar 14 2015

Extensions

More terms from Michael Somos
Typo in b-file description fixed by David Applegate, Sep 05 2009

A063966 Number of Abelian groups of order <= n.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 8, 11, 13, 14, 15, 17, 18, 19, 20, 25, 26, 28, 29, 31, 32, 33, 34, 37, 39, 40, 43, 45, 46, 47, 48, 55, 56, 57, 58, 62, 63, 64, 65, 68, 69, 70, 71, 73, 75, 76, 77, 82, 84, 86, 87, 89, 90, 93, 94, 97, 98, 99, 100, 102, 103, 104, 106, 117, 118, 119, 120, 122
Offset: 1

Views

Author

Ahmed Fares (ahmedfares(AT)my-deja.com), Sep 04 2001

Keywords

Links

Crossrefs

Partial sums of A000688.
Cf. A063756.
Cf. A021002, A084892, A084893.

Programs

  • Maple
    with(combinat): readlib(ifactors): total := 0: for n from 1 to 100 do ans := 1: for i from 1 to nops(ifactors(n)[2]) do ans := ans*numbpart(ifactors(n)[2][i][2]) od: printf(`%d,`,total+ans): total := total+ans: od:
  • Mathematica
    Accumulate[Table[FiniteAbelianGroupCount[n], {n, 1, 200}]] (* Geoffrey Critzer, Dec 28 2014 *)

Formula

a(n) ~ c * n, where c = A021002 = Product_{k>=2} zeta(k). - Vaclav Kotesovec, Oct 26 2019
More accurately, a(n) = A021002 * n + A084892 * n^(1/2) + A084893 * n^(1/3) + O(n^(50/199 + eps)), where eps>0 is arbitrarily small (Liu, 1993). - Amiram Eldar, Sep 23 2023

Extensions

More terms from James Sellers, Sep 26 2001

A118581 Number of nonisomorphic semigroups of order <= n.

Original entry on oeis.org

1, 2, 7, 31, 219, 2134, 30768, 1658440, 3685688857, 105981863625149
Offset: 0

Views

Author

Jonathan Vos Post, May 07 2006

Keywords

Comments

Semigroup analog of A063756 Number of groups of order <= n. a semigroup is an algebraic structure consisting of a set S closed under an associative binary operation (and thus is an associative groupoid). Some sources require that a semigroup have an identity element (in which case semigroups are identical to monoids). Not all sources agree that S should be nonempty. This sequence assumes that a semigroup may be empty and need not have an identity.

Examples

			a(7) = 1658440 = 1 + 1 + 5 + 24 + 188 + 1915 + 28634 + 1627672.

Crossrefs

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A063756, A079173.

Formula

a(n) = Sum_{i=0..n} A027851(i). a(n) = Sum_{i=0..n} (2*A001423(i) - A029851(i)).

Extensions

a(8)-a(9) (using A027851) from Giovanni Resta, Jun 16 2016

A118601 Partial sums of A058129.

Original entry on oeis.org

1, 3, 10, 45, 273, 2510, 34069, 1703066
Offset: 1

Views

Author

Jonathan Vos Post, May 08 2006

Keywords

Crossrefs

Cf. A001329, A001423, A001426, A023814, A027851, A029851, A058108, A058132, A058133, A063756, A079173, A118581.

Formula

a(n) = SUM[i=1..n] A058129(i). a(n) = SUM[i=1..n] (2*A058133(i) - A058132(i)).

Extensions

One more term from Jonathan Vos Post, Jul 20 2009
Edited by N. J. A. Sloane, Jul 25 2009

A173666 Partial sums of number of supersolvable groups of order n (A066083).

Original entry on oeis.org

1, 2, 3, 5, 6, 8, 9, 14, 16, 18, 19, 23, 24, 26, 27, 41, 42, 47, 48, 53, 55, 57, 58, 70, 72, 74, 79, 83, 84, 88, 89, 140, 141, 143, 144, 155, 156, 158, 160, 174, 175, 181, 182, 186, 188, 190, 191, 233, 235, 240, 241, 246, 247, 262, 264, 276, 278, 280, 281, 292, 293, 295, 299, 566, 567, 571, 572, 577, 578, 582, 583, 620, 621, 623, 625, 629, 630, 636, 637, 688
Offset: 1

Views

Author

Jonathan Vos Post, Nov 24 2010

Keywords

Comments

Number of supersolvable groups of order <= n. Diverges from A063756 after n=11. The subsequence of primes in this sequence begins: 2, 3, 5, 19, 23, 41, 47, 53, 79, 83, 89, 181, 191, 233, 241, 281, 293, 571, 577. The subsequence of perfect powers in this sequence begins: 1, 8, 9, 16, 27, 144, 625.

Crossrefs

Cf. A000001, A063756, A066085, A066083.

Programs

Formula

a(n) = Sum_{i=1..n} A066083 = Sum_{i=1..n} number of finite groups of order i for which the index of any maximal subgroup is prime.
Showing 1-6 of 6 results.

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