%I A000001 M0098 N0035
%S A000001 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,
%T A000001 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,
%U A000001 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
%N A000001 Number of groups of order n.
%C A000001 Also, number of nonisomorphic subgroups of order n in symmetric group 
               S_n. - Lekraj Beedassy (blekraj(AT)yahoo.com), Dec 16 2004
%D A000001 H. A. Bender, A determination of the groups of order p^5, Ann. of Math. 
               (2) 29, pp. 61-72 (1927).
%D A000001 H.-U. Besche and B. Eick, Construction of Finite Groups, Journal of Symbolic 
               Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
%D A000001 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.
%D A000001 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.
%D A000001 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
%D A000001 J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
%D A000001 H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete 
               Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
%D A000001 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
%D A000001 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.
%D A000001 M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, 
               NY, 1964.
%D A000001 Otto Holder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 
               43 pp. 301-412 (1893).
%D A000001 G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 
               52 (1930), 617-634.
%D A000001 D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, 
               p. 481.
%D A000001 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.
%D A000001 E. Rodemich, The groups of order 128. J. Algebra 67 (1980), no. 1, 129-142.
%D A000001 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000001 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A000001 M. Wild, The groups of order 16 made easy, Amer. Math. Monthly, 112 (No. 
               1, 2005), 20-31.
%H A000001 H.-U. Besche and Ivan Panchenko, <a href="b000001.txt">Table of n, a(n) 
               for n = 1..2047</a> [Terms 1 through 2015 copied from Small Groups 
               Library mentioned below. Terms 2016 - 2047 added by Ivan Panchenko, 
               Aug 29 2009]
%H A000001 H.-U. Besche, <a href="http://www-public.tu-bs.de:8080/~hubesche/small.html">
               The Small Groups Library</a> [gives 2000 terms]
%H A000001 H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.ams.org/
               era/2001-07-01/S1079-6762-01-00087-7/home.html">The groups of order 
               at most 2000</a>, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 
               1-4.
%H A000001 H. Bottomley, <a href="a1.gif">Illustration of initial terms</a>
%H A000001 J. H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/
               ~obrien/research/gnu.pdf">Counting groups: gnus, moas and other exotica</
               a>.
%H A000001 Ed Pegg Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_12_08_03.html">
               Illustration of initial terms</a>
%H A000001 Gordon Royle, <a href="http://www.cs.uwa.edu.au/~gordon/remote/cubcay/
               index.html">Numbers of Small Groups</a>
%H A000001 D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/95/numgrps">
               Asymptotics</a>.
%H A000001 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               FiniteGroup.html">Link to a section of The World of Mathematics.</
               a>
%H A000001 G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~algebra~smallgroup.en.html">
               SmallGroup</a>
%H A000001 <a href="Sindx_Gre.html#groups">Index entries for sequences related to 
               groups</a>
%H A000001 <a href="Sindx_Cor.html#core">Index entries for "core" sequences</a>
%F A000001 Formulae from Mitch Harris (Harris.Mitchell(AT)mgh.harvard.edu), Oct 
               25 2006
%F A000001 (Start) For p, q, r primes:
%F A000001 a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
%F A000001 a(p^5) = 61 + 2p + 2gcd(p-1,3) + gcd(p-1,4), p>=5, a(2^5)=51, a(3^5)=67.
%F A000001 a(p^e) ~ p^((2/27)e^3 + O(e^(8/3)))
%F A000001 a(pq) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
%F A000001 a(pq^2) = one of the following:
%F A000001 * 5, p=2, q odd,
%F A000001 * (p+9)/2, q=1 mod p, p odd,
%F A000001 * 5, p=3, q=2,
%F A000001 * 3, q = -1 mod p, p and q odd.
%F A000001 * 4, p=1 mod q, p > 3, p != 1 mod q^2
%F A000001 * 5, p=1 mod q^2
%F A000001 * 2, q != +/-1 mod p and p != 1 mod q,
%F A000001 a(pqr) (p < q < r) = one of the following:
%F A000001 * q==1 mod p r==1 mod p r==1 mod q a(pqr)
%F A000001 * No..........No..........No..........1
%F A000001 * No..........No..........Yes.........2
%F A000001 * No..........Yes.........No..........2
%F A000001 * No..........Yes.........Yes.........4
%F A000001 * Yes.........No..........No..........2
%F A000001 * Yes.........No..........Yes.........3
%F A000001 * Yes.........Yes.........No..........p+2
%F A000001 * Yes.........Yes.........Yes.........p+4 (table from Derek Holt) (End)
%o A000001 (MAGMA) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] 
               ]; (from John Cannon, Dec 23 2006)
%Y A000001 The main sequences concerned with group theory are A000001 (this one), 
               A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, 
               A005432.
%Y A000001 Cf. A046058, A001493, A023675, A023676. A003277 gives n for which A000001(n) 
               = 1.
%Y A000001 Sequence in context: A119569 A066083 A128644 this_sequence A146002 A109087 
               A102048
%Y A000001 Adjacent sequences:    this_sequence A000002 A000003 A000004
%K A000001 nonn,core,nice
%O A000001 1,4
%A A000001 N. J. A. Sloane (njas(AT)research.att.com).
%E A000001 More terms from Michael Somos
%E A000001 Typo in b-file description fixed by David Applegate (david(AT)research.att.com), 
               Sep 05 2009