|
%I M1470 N0581 #89 Mar 10 2024 04:02:34
%S 1,1,2,5,14,51,267,2328,56092,10494213,49487367289
%N Number of groups of order 2^n.
%D James Gleick, Faster, Vintage Books, NY, 2000 (see pp. 259-261).
%D M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
%D M. F. Newman, Groups of prime-power order (1990). In Groups—Canberra 1989 (pp. 49-62). Springer, Berlin, Heidelberg. See Table 1.
%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 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).
%H T. Applebaum, J. Clikeman, J. A. Davis, J. F. Dillon, J. Jedwab, T. Rabbani, K. Smith, and W. Yolland, <a href="https://doi.org/10.2140/ant.2023.17.359">Constructions of difference sets in nonabelian 2-groups</a>, Alg. Num. Theor. (2023) Vol. 17, No. 2. 359-396. See p. 396.
%H Hans Ulrich Besche and Bettina Eick, <a href="https://doi.org/10.1006/jsco.1998.0258">Construction of finite groups</a>, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
%H Hans Ulrich Besche and Bettina Eick, <a href="http://www.icm.tu-bs.de/~beick/publ/1000.ps">The groups of order at most 1000 except 512 and 768</a>, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 405-413.
%H Hans Ulrich Besche, Bettina Eick and E. A. O'Brien, <a href="https://doi.org/10.1090/S1079-6762-01-00087-7">The groups of order at most 2000</a>, Electron. Res. Announc. Amer. Math. Soc. 7 (2001), 1-4.
%H David Burrell, <a href="https://doi.org/10.1080/00927872.2021.2006680">On the number of groups of order 1024</a>, Communications in Algebra, 2021, 1-3.
%H Hans Ulrich Besche, <a href="http://www.icm.tu-bs.de/ag_algebra/software/small/small.html">The Small Groups library</a>
%H Bettina Eick and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/count-pgroups.pdf">Enumerating p-groups</a>. Group theory. J. Austral. Math. Soc. Ser. A 67 (1999), no. 2, 191-205.
%H Richard K. Guy, <a href="/A005347/a005347.pdf">The Second Strong Law of Small Numbers</a>, Math. Mag, 63 (1990), no. 1, 3-20. [Annotated scanned copy]
%H Rodney James and John Cannon, <a href="http://dx.doi.org/10.1090/S0025-5718-1969-0238953-8">Computation of isomorphism classes of p-groups</a>, Mathematics of Computation 23.105 (1969): 135-140.
%H Rodney James, M. F. Newman, and E. A. O'Brien, <a href="https://doi.org/10.1016/0021-8693(90)90244-I">The Groups of Order 128</a>, J. Algebra 129, 136-158, 1990.
%H G. A. Miller, <a href="http://www.jstor.org/stable/2370630">Determination of all the groups of order 64</a>, Amer. J. Math., 52 (1930), 617-634.
%H E. A. O'Brien, <a href="https://doi.org/10.1016/0021-8693(91)90261-6">The Groups of Order 256</a> J. Algebra 143, 219-235, 1991.
%H Eugene Rodemich, <a href="https://doi.org/10.1016/0021-8693(80)90312-9">The groups of order 128</a>, J. Algebra 67 (1980), no. 1, 129-142.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group</a>.
%H Marcel Wild, <a href="https://doi.org/10.2307/30037381">The groups of order 16 made easy</a>, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>.
%F a(n) = 2^((2/27)n^3 + O(n^(8/3))).
%F a(n) = A000001(2^n). - _Amiram Eldar_, Mar 10 2024
%e G.f. = 1 + x + 2*x^2 + 5*x^3 + 14*x^4 + 51*x^5 + 267*x^6 + 2328*x^7 + ...
%p seq(GroupTheory:--NumGroups(2^n),n=0..10); # _Robert Israel_, Oct 15 2017
%t Join[{1}, FiniteGroupCount[2^Range[10]]] (* _Vincenzo Librandi_, Mar 28 2018 *)
%o (GAP) A000679 := List([0..8],n -> NumberSmallGroups(2^n)); # _Muniru A Asiru_, Oct 15 2017
%Y Cf. A000001, A046058.
%K nonn,hard,more,nice
%O 0,3
%A _N. J. A. Sloane_
%E a(9) and a(10) found by Eamonn O'Brien
%E a(10) corrected by _David Burrell_, Jun 06 2022
|
