A053403 -id:A053403 - cp's OEIS frontend

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

Eric W. Weisstein

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)

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

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

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

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

A240007 Smallest k such that the number of groups of order k is equal to prime(n), or 0 if no such k exists.

Original entry on oeis.org

4, 75, 8, 375, 140, 56, 675, 1029, 294, 1380, 0, 180, 420, 112, 120, 656, 6875, 312, 243, 3660, 0, 3612, 0, 4140, 6498, 0, 0, 0, 0, 810, 0, 1260, 792, 0, 0, 0, 0, 0, 1936, 0, 1456, 1320, 0, 0, 144, 1000, 1368, 0, 1404, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Michel Lagneau, Mar 30 2014

nonn

Smallest k such that A000001(k) = prime(n), or 0 if no such k exists.
It seems that there is no order for which the number of groups is 31, 59, 71, 73, 79, 83, 89, 97, 101, 103, 109, 127, 139,...
Above comment is incorrect. According to the Conway article, every n <= 10000000 is the number of groups of order k for some k. So all the 0 entries above are wrong, but we do not necessarily know the true value. - Eric M. Schmidt, Sep 14 2014

			a(6)= 56 because prime(6) = 13 => there exists 13 groups of order 56.

John H. Conway, Heiko Dietrich and E. A. O'Brien, Counting groups: gnus, moas and other exotica, The Mathematical Intelligencer, Spring 2008, Volume 30, Issue 2, pp 6-15, doi:10.1007/BF02985731.

Cf. A000001, A046057, A053403.

lst={};Do[k=1;While[!FiniteGroupCount[k]==Prime[n],k++];If[k==2048,AppendTo[lst,0],AppendTo[lst,k]],{n,1,70}];lst

Values for 59, 71, 79, 89, and 97 filled in from Conway link by Eric M. Schmidt, Sep 14 2014

cp's OEIS Frontend

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

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