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A000001 Number of groups of order n.
(Formerly M0098 N0035)
114
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,4

COMMENTS

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy, Dec 16 2004

Also, number of nonisomorphic primitives of the combinatorial species Lin[n-1]. - Nicolae Boicu, Apr 29 2011

The record values are (A046058): 1, 2, 5, 14, 15, 51, 52, 267, 2328, 56092, 10494213, 49487365422, ..., and they appear at positions (A046059): 1, 4, 8, 16, 24, 32, 48, 64, 128, 256, 512, 1024, .... Robert G. Wilson v, Oct 12 2012

REFERENCES

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' pgs 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', John 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

H.-U. Besche and Ivan Panchenko, Table of n, a(n) for n = 1..2047 [Terms 1 through 2015 copied from Small Groups Library mentioned below. Terms 2016 - 2047 added by Ivan Panchenko, Aug 29 2009]

H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).

Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.

Hans Ulrich Besche and Bettina Eick, The groups of order at most 1000 except 512 and 768, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.

H. U. Besche, B. Eick and E. A. O'Brien, The groups of order at most 2000, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.

H. U. Besche, B. Eick and E. A. O'Brien, The Small Groups Library

H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order

H.-U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.

H. Bottomley, Illustration of initial terms

J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.

Otto Hölder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893).

G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.

Ed Pegg Jr., Illustration of initial terms [broken link?]

D. S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland.

E. Rodemich, The groups of order 128, J. Algebra 67 (1980), no. 1, 129-142.

Gordon Royle, Numbers of Small Groups [broken link?]

D. Rusin, Asymptotics.

Eric Weisstein's World of Mathematics, Finite Group

Wikipedia, Finite group

M. Wild, The groups of order 16 made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.

Gang Xiao, SmallGroup

Index entries for sequences related to groups

Index entries for "core" sequences

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 + 2p + 2gcd(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(pq) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)

a(pq^2) = one of the following:

* 5, p=2, q odd,

* (p+9)/2, q=1 mod p, p odd,

* 5, p=3, q=2,

* 3, q = -1 mod p, p and q odd.

* 4, p=1 mod q, p > 3, p != 1 mod q^2

* 5, p=1 mod q^2

* 2, q != +/-1 mod p and p != 1 mod q,

a(pqr) (p < q < r) = one of the following:

* q==1 mod p r==1 mod p r==1 mod q a(pqr)

* 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)

MAPLE

GroupTheory:-NumGroups( n )

MATHEMATICA

FiniteGroupCount[Range[100]] (* Harvey P. Dale, Jan 29 2013 *)

a[ n_] := If[ n < 1, 0, FiniteGroupCount @ n]; (* Michael Somos, May 28 2014 *)

PROG

(MAGMA) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; // John Cannon, Dec 23 2006

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, A023675, A023676. A003277 gives n for which A000001(n) = 1.

Sequence in context: A066083 A128644 A201733 * A172133 A146002 A109087

Adjacent sequences:  * A000002 A000003 A000004 A000005

KEYWORD

nonn,core,nice,hard

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from Michael Somos

Typo in b-file description fixed by David Applegate, Sep 05 2009

Corrected a broken link. - N. J. A. Sloane, May 10 2012

STATUS

approved

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Last modified July 26 13:19 EDT 2014. Contains 244946 sequences.