1,4

Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 16 2004

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

H.-U. Besche and B. Eick, Construction of Finite Groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.

H.-U. Besche and B. 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, A Millenium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.

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.

Otto Holder, 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.

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.

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

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

H.-U. Besche, Table of n, a(n) for n = 1..2015 [Copied from Small Groups Library mentioned below]

H.-U. Besche, The Small Groups Library [gives 2000 terms]

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. Bottomley, Illustration of initial terms

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

Ed Pegg Jr., Illustration of initial terms

Gordon Royle, Numbers of Small Groups

D. Rusin, Asymptotics.

Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.

G. Xiao, SmallGroup

Index entries for sequences related to groups

Index entries for "core" sequences

Formulae from Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), 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)

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

The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432.

Cf. A046058, A001493, A023675, A023676. A003277 gives n for which A000001(n) = 1.

Adjacent sequences: this_sequence A000002 A000003 A000004

Sequence in context: A119569 A066083 A128644 this_sequence A146002 A109087 A102048

nonn,core,nice

N. J. A. Sloane (njas(AT)research.att.com).

More terms from Michael Somos