A128694 - OEIS

%I #7 Sep 08 2022 08:45:30

%S 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 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 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 Number of groups of order A128693(n).

%C Number of groups for orders of form 3^k*p, where 1 <= k <= 6 and p is a prime different from 3.

%C 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 Klaus Brockhaus, <a href="/A128694/b128694.txt">Table of n, a(n) for n=1..10000</a>

%H MAGMA Documentation, <a href="http://magma.maths.usyd.edu.au/magma/htmlhelp/text404.htm">Database of Small Groups</a>

%F a(n) = A000001(A128693(n)).

%e A128693(9) = 54 and there are 15 groups of order 54 (A000001(54) = 15), hence a(9) = 15.

%o (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 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 nonn

%O 1,1

%A _Klaus Brockhaus_, Mar 26 2007