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A128694 Number of groups of order A128693(n).

%I A128694 #11 Mar 15 2025 10:45:14
%S A128694 2,1,5,2,1,2,2,1,15,2,4,1,1,2,2,2,4,1,2,5,1,2,1,55,5,1,2,13,2,2,1,2,2,
%T A128694 1,2,1,4,2,5,1,2,1,2,5,1,14,2,2,4,1,16,1,2,2,1,2,5,2,2,261,2,1,15,1,2,
%U A128694 1,2,4,49,1,2,1,2,4,5,2,2,5,2,1,2,1,4,1,2,2,1,1,5,1,2,1,2,2,13,1,2,4,1,15,2
%N A128694 Number of groups of order A128693(n).
%C A128694 Number of groups for orders of form 3^k*p, where 1 <= k <= 6 and p is a prime different from 3.
%C A128694 The groups of these orders (up to A128693(84005521) = 3221225379 in version V2.13-4) form a class contained in the Small Groups Library of MAGMA.
%H A128694 Klaus Brockhaus, <a href="/A128694/b128694.txt">Table of n, a(n) for n=1..10000</a>
%H A128694 Magma Computational Algebra System, <a href="https://magma.maths.usyd.edu.au/magma/handbook/">Documentation</a>, see Database of Small Groups.
%F A128694 a(n) = A000001(A128693(n)).
%e A128694 A128693(9) = 54 and there are 15 groups of order 54 (A000001(54) = 15), hence a(9) = 15.
%o A128694 (Magma) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n): n in [ h: h in [1..910] | #t eq 2 and ((t[1, 1] eq 2 and t[1, 2] eq 1 and t[2, 1] eq 3 and t[2, 2] le 6) or (t[1, 1] eq 3 and t[1, 2] le 6 and t[2, 2] eq 1)) where t is Factorization(h) ] ];
%Y A128694 Cf. A000001 (number of groups of order n), A128693 (numbers of form 3^k*p, 1<=k<=6, p!=3 prime), A128604 (number of groups for orders that divide p^6, p prime), A128644 (number of groups for orders that have at most 3 prime factors), A128645 (number of groups for orders of form 2^k*p, 1<=k<=8 and p>2 prime).
%K A128694 nonn
%O A128694 1,1
%A A128694 _Klaus Brockhaus_, Mar 26 2007