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%I #16 Sep 15 2014 04:11:03
%S 4,75,8,375,140,56,675,1029,294,1380,0,180,420,112,120,656,6875,312,
%T 243,3660,0,3612,0,4140,6498,0,0,0,0,810,0,1260,792,0,0,0,0,0,1936,0,
%U 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
%N Smallest k such that the number of groups of order k is equal to prime(n), or 0 if no such k exists.
%C Smallest k such that A000001(k) = prime(n), or 0 if no such k exists.
%C 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,...
%C 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
%H John H. Conway, Heiko Dietrich and E. A. O'Brien, <a href="http://www.math.auckland.ac.nz/~obrien/research/gnu.pdf">Counting groups: gnus, moas and other exotica</a>, The Mathematical Intelligencer, Spring 2008, Volume 30, Issue 2, pp 6-15, doi:10.1007/BF02985731.
%e a(6)= 56 because prime(6) = 13 => there exists 13 groups of order 56.
%t lst={};Do[k=1;While[!FiniteGroupCount[k]==Prime[n],k++];If[k==2048,AppendTo[lst,0],AppendTo[lst,k]],{n,1,70}];lst
%Y Cf. A000001, A046057, A053403.
%K nonn
%O 1,1
%A _Michel Lagneau_, Mar 30 2014
%E Values for 59, 71, 79, 89, and 97 filled in from Conway link by _Eric M. Schmidt_, Sep 14 2014
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