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

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

Author

Michel Lagneau, Mar 30 2014

Keywords

• nonn

Comments

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

Examples

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

Links

• 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.

Crossrefs

Cf. A000001, A046057, A053403.

Programs

Mathematica

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

Extensions

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