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%I #78 Jan 17 2022 11:27:09
%S 1,4,75,28,8,42,375,510,308,90,140,88,56,16,24,100,675,156,1029,820,
%T 1875,6321,294,546,2450,2550,1210,2156,1380,270,11774,630
%N Smallest order m > 0 for which there are n nonisomorphic finite groups of order m, or 0 if no such order exists.
%C R. Keith Dennis conjectures that there are no 0's in this sequence. See A053403 for details.
%C In (John H. Conway, Heiko Dietrich and E. A. O'Brien, 2008), m is called the "minimal order attaining n" and is denoted by moa(n). - _Daniel Forgues_, Feb 15 2017
%C a(33) > 30500. - _Muniru A Asiru_, Nov 15 2017
%C From _Jorge R. F. F. Lopes_, Jan 07 2022: (Start)
%C The following values taken from the Max Horn website are improvements over those given in the Conway-Dietrich-O'Brien table (see Links):
%C a(58) = 3591, a(59) = 6328, a(63) = 2025, a(73) = 24003, a(74) = 25250, a(78) = 12750, a(90) = 2970, a(91) = 2058, a(92) = 15092. (End)
%D J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.
%H H. U. Besche, <a href="https://www.gap-system.org/Packages/sgl.html">The Small Groups library</a>
%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="https://doi.org/10.1006/jsco.1998.0259">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 John 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>, The Mathematical Intelligencer, Volume 30, Issue 2 (2008), pp 6-15, DOI:<a href="https://doi.org/10.1007/BF02985731">10.1007/BF02985731</a>.
%H John H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="/A046057/a046057_1.txt">Table of n, a(n) for n = 1..100</a> (with some question marks) [contains some errors: see comment from _Jorge R. F. F. Lopes_].
%H Steven Finch, <a href="https://doi.org/10.1007/s00283-021-10060-2">The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane</a>, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.
%H Max Horn, <a href="https://groups.quendi.de/">Numbers of isomorphism types of finite groups of given order</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FiniteGroup.html">Finite Group.</a>
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%Y Cf. A000001, A046056, A046058, A046059, A053403.
%K nonn,hard,more,nice
%O 1,2
%A _Eric W. Weisstein_
%E More terms from Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Nov 04 2003
%E a(20) corrected by _N. J. A. Sloane_, Jan 21 2004
%E More terms from _N. J. A. Sloane_, Oct 03 2008, from the John H. Conway, Heiko Dietrich and E. A. O'Brien article.
%E a(31)-a(32) from _Muniru A Asiru_, Nov 15 2017