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%I M0098 N0035
%S 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 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 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 Number of groups of order n.
%C Also, number of nonisomorphic subgroups of order n in symmetric group S_n. - _Lekraj Beedassy_, Dec 16 2004
%C Also, number of nonisomorphic primitives of the combinatorial species Lin[n-1] - _Nicolae Boicu_, April 29 2011
%C 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
%D H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
%D 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 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 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.
%D L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 302, #35.
%D J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
%D 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 CRC Standard Mathematical Tables and Formulae, 30th ed. 1996, p. 150.
%D 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 M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
%D Otto Holder, Die Gruppen der Ordnungen p^3, pq^2, pqr, p^4, Math. Ann. 43 pp. 301-412 (1893).
%D G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
%D D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
%D 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 D. S. Rajan, The equations D^kY=X^n in combinatorial species, Discrete Mathematics 118 (1993) 197-206 North-Holland.
%D E. Rodemich, The groups of order 128. J. Algebra 67 (1980), no. 1, 129-142.
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D M. Wild, The groups of order 16 made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
%H H.-U. Besche and Ivan Panchenko, <a href="/A000001/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 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 H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.icm.tu-bs.de/ag_algebra/software/small/">The Small Groups Library</a>
%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://www.icm.tu-bs.de/ag_algebra/software/small/number.html">Number of isomorphism types of finite groups of given order</a>
%H H. Bottomley, <a href="/A000001/a000001.gif">Illustration of initial terms</a>
%H 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 Ed Pegg Jr., <a href="http://www.maa.org/editorial/mathgames/mathgames_12_08_03.html">Illustration of initial terms</a>
%H Gordon Royle, <a href="http://www.cs.uwa.edu.au/~gordon/remote/cubcay/index.html">Numbers of Small Groups</a>
%H D. Rusin, <a href="http://www.math.niu.edu/~rusin/known-math/95/numgrps">Asymptotics</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Finite_group">Finite group</a>
%H G. Xiao, <a href="http://wims.unice.fr/~wims/en_tool~algebra~smallgroup.en.html">SmallGroup</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%H <a href="/index/Cor#core">Index entries for "core" sequences</a>
%F Formulae from _Mitch Harris_, Oct 25 2006
%F (Start) For p, q, r primes:
%F a(p) = 1, a(p^2) = 2, a(p^3) = 5, a(p^4) = 14, if p = 2, otherwise 15.
%F a(p^5) = 61 + 2p + 2gcd(p-1,3) + gcd(p-1,4), p>=5, a(2^5)=51, a(3^5)=67.
%F a(p^e) ~ p^((2/27)e^3 + O(e^(8/3)))
%F a(pq) = 1 if gcd(p,q-1) = 1, 2 if gcd(p,q-1) = p. (p < q)
%F a(pq^2) = one of the following:
%F * 5, p=2, q odd,
%F * (p+9)/2, q=1 mod p, p odd,
%F * 5, p=3, q=2,
%F * 3, q = -1 mod p, p and q odd.
%F * 4, p=1 mod q, p > 3, p != 1 mod q^2
%F * 5, p=1 mod q^2
%F * 2, q != +/-1 mod p and p != 1 mod q,
%F a(pqr) (p < q < r) = one of the following:
%F * q==1 mod p r==1 mod p r==1 mod q a(pqr)
%F * No..........No..........No..........1
%F * No..........No..........Yes.........2
%F * No..........Yes.........No..........2
%F * No..........Yes.........Yes.........4
%F * Yes.........No..........No..........2
%F * Yes.........No..........Yes.........3
%F * Yes.........Yes.........No..........p+2
%F * Yes.........Yes.........Yes.........p+4 (table from Derek Holt) (End)
%t aa = {}; Do[AppendTo[aa, FiniteGroupCount[n]], {n, 1, 50}]; aa (*Artur Jasinski*) [From _Artur Jasinski_, May 29 2010]
%t FiniteGroupCount[Range[100]] (* _Harvey P. Dale_, Jan 29 2013 *)
%o (MAGMA) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [1..1000] ]; (from John Cannon, Dec 23 2006)
%Y The main sequences concerned with group theory are A000001 (this one), A000679, A001034, A001228, A005180, A000019, A000637, A000638, A002106, A005432, A000688, A060689, A051532.
%Y Cf. A046058, A023675, A023676. A003277 gives n for which A000001(n) = 1.
%K nonn,core,nice,hard
%O 1,4
%A _N. J. A. Sloane_.
%E More terms from Michael Somos
%E Typo in b-file description fixed by _David Applegate_, Sep 05 2009
%E Corrected a broken link. - _N. J. A. Sloane_, May 10 2012
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