A208663 Non-Abelian numbers: n such that A000001(n)/A000688(n) is a new record.

1, 6, 12, 16, 24, 32, 48, 64, 96, 128, 256, 512, 1024, 2048

Offset: 1

Author

Ben Branman, Feb 29 2012

Keywords

• nonn
• hard
• more

Examples

			For a(n)=12, there are 2 Abelian groups and 3 nonabelian groups, so the ratio A000001(12)/A000688(12)=5/2=2.5, which beats the previous record of 2, so 12 is in the sequence.
		

References

• H. A. Bender, A determination of the groups of order p^5, Ann. of Math. (2) 29, pp. 61-72 (1927).
• H. U. Besche and B. Eick, Construction of Finite Groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
• 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.
• 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.
• H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984, p. 134.
• M. Hall, Jr. and J. K. Senior, The Groups of Order 2^n (n <= 6). Macmillan, NY, 1964.
• G. A. Miller, Determination of all the groups of order 64, Amer. J. Math., 52 (1930), 617-634.
• D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.24, p. 481.
• 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.
• E. Rodemich, The groups of order 128. J. Algebra 67 (1980), no. 1, 129-142.

Links

• J. H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica.
• Eric Weisstein's World of Mathematics, Finite Group
• M. Wild, The groups of order sixteen made easy, Amer. Math. Monthly, 112 (No. 1, 2005), 20-31.
• Index entries for sequences related to groups

Crossrefs

Cf. A000001, A000688, A060689, A051532.

Programs

Mathematica

s = {1}; a = 1; Do[b = FiniteGroupCount[n]/FiniteAbelianGroupCount[n];
  If[b > a, a = b; AppendTo[s, n]], {n, 1, 2047}]; s

Extensions

a(14) from Eric M. Schmidt, Aug 02 2012