A046057 Smallest order m > 0 for which there are n nonisomorphic finite groups of order m, or 0 if no such order exists.

1, 4, 75, 28, 8, 42, 375, 510, 308, 90, 140, 88, 56, 16, 24, 100, 675, 156, 1029, 820, 1875, 6321, 294, 546, 2450, 2550, 1210, 2156, 1380, 270, 11774, 630

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• hard
• more
• nice

Comments

R. Keith Dennis conjectures that there are no 0's in this sequence. See A053403 for details.

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

a(33) > 30500. - Muniru A Asiru, Nov 15 2017

From Jorge R. F. F. Lopes, Jan 07 2022: (Start)

The following values taken from the Max Horn website are improvements over those given in the Conway-Dietrich-O'Brien table (see Links):

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)

References

• J. H. Conway et al., The Symmetries of Things, Peters, 2008, p. 209.

Links

• H. U. Besche, The Small Groups library
• Hans Ulrich Besche and Bettina Eick, Construction of finite groups, Journal of Symbolic Computation, Vol. 27, No. 4, Apr 15 1999, pp. 387-404.
• Hans Ulrich Besche and Bettina 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.
• John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, The Mathematical Intelligencer, Volume 30, Issue 2 (2008), pp 6-15, DOI:10.1007/BF02985731.
• John H. Conway, Heiko Dietrich and E. A. O'Brien, Table of n, a(n) for n = 1..100 (with some question marks) [contains some errors: see comment from _Jorge R. F. F. Lopes_].
• Steven Finch, The On-Line Encyclopedia of Integer Sequences, founded in 1964 by N. J. A. Sloane, A Tribute to John Horton Conway, The Mathematical Intelligencer (2021) Vol. 43, 146-147.
• Max Horn, Numbers of isomorphism types of finite groups of given order
• Eric Weisstein's World of Mathematics, Finite Group.
• Index entries for sequences related to groups

Crossrefs

Cf. A000001, A046056, A046058, A046059, A053403.

Extensions

More terms from Victoria A. Sapko (vsapko(AT)canes.gsw.edu), Nov 04 2003

a(20) corrected by N. J. A. Sloane, Jan 21 2004

More terms from N. J. A. Sloane, Oct 03 2008, from the John H. Conway, Heiko Dietrich and E. A. O'Brien article.

a(31)-a(32) from Muniru A Asiru, Nov 15 2017