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%I #38 Sep 01 2023 11:27:43
%S 1,1,2,5,15,67,504,9310,1396077,5937876645
%N Number of groups of order 3^n.
%D G. Bagnera, La composizione dei Gruppi finiti il cui grado e la quinta potenza di un numero primo, Ann. Mat. Pura Appl. (3), 1 (1898), 137-228.
%D Hans Ulrich Besche, Bettina Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, International Journal of Algebra and Computation, Vol. 12, No 5 (2002), 623-644.
%D W. Burnside, Theory of Groups of Finite Order, Dover, NY, 1955.
%D Marcus du Sautoy, Symmetry: A Journey into the Patterns of Nature, HarperCollins, 2008, p. 96.
%H David Burrell, <a href="https://doi.org/10.1080/00927872.2023.2169706">The number of p-groups of order 19,683 and new lists of p-groups</a>, Communications in Algebra, Vol. 51 - Issue 6 (2023), 2673-2679.
%H Heiko Dietrich, <a href="http://users.monash.edu/~heikod/icts2016/CPGmain.pdf">Computational aspects of finite p-groups</a>, 2016.
%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 M. F. Newman, E. A. O'Brien and M. R. Vaughan-Lee, <a href="http://dx.doi.org/10.1016/j.jalgebra.2003.11.012">Groups and nilpotent Lie rings whose order is the sixth power of a prime</a>, J. Algebra, 278 (2004), 383-401.
%H E. A. O'Brien and M. R. Vaughan-Lee, <a href="http://dx.doi.org/10.1016/j.jalgebra.2005.01.019">The groups of order p^7 for odd prime p</a>, J. Algebra 292, 243-258, 2005. [_David Radcliffe_, Feb 24 2010]
%H Michael Vaughan-Lee, <a href="https://dx.doi.org/10.1365/s13291-012-0039-x">Graham Higman’s PORC Conjecture</a>, Jahresbericht der Deutschen Mathematiker-Vereinigung Vol. 114 (2012), 89-16.
%H Michael Vaughan-Lee, <a href="http://dx.doi.org/10.22108/ijgt.2015.5758">Groups of order p^8 and exponent p</a>, International Journal of Group Theory Vol. 4 (2015), 25-42.
%H Brett Edward Witty, <a href="https://www.brettwitty.net/pages/phd.html">Enumeration of groups of prime-power order</a>, PhD thesis, 2006.
%F a(n) = A000001(3^n).
%e G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 67*x^5 + 504*x^6 + 9310*x^7 + ...
%p with(GroupTheory): seq(NumGroups(3^n),n=0..8); # _Muniru A Asiru_, Oct 17 2018
%o (GAP) A090091 := List([0..7],n -> NumberSmallGroups(3^n)); # _Muniru A Asiru_, Oct 15 2017
%Y Cf. A000001, A000679, A090130, A090140.
%K nonn,hard,more
%O 0,3
%A Eamonn O'Brien (obrien(AT)math.auckland.ac.nz), Jan 22 2004
%E a(7) from _David Radcliffe_, Feb 24 2010
%E a(8) from _Muniru A Asiru_, Oct 17 2018
%E a(9) from _David Burrell_, Sep 01 2023
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