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Number of groups of order A128703(n).
3

%I #11 Sep 20 2024 05:42:47

%S 2,1,1,5,2,1,3,1,1,1,1,2,2,1,2,1,1,15,1,4,1,2,2,1,2,1,7,1,1,2,1,2,1,2,

%T 1,1,1,1,2,1,2,1,1,2,1,2,4,1,1,1,1,5,1,2,2,2,1,1,1,4,2,2,1,1,1,1,2,1,

%U 2,55,2,1,1,2,1,2,15,1,2,1,1,2,4,1,2,1,1,5,2,2,1,1,1,1,4,1,2,1,1,21,1,1,1,2

%N Number of groups of order A128703(n).

%C Number of groups for orders of form 5^k*p, where 1 <= k <= 5 and p is a prime different from 5.

%C The groups of these orders (up to A128703(69556991) = 5368708945 in version V2.13-4) form a class contained in the Small Groups Library of MAGMA.

%H Klaus Brockhaus, <a href="/A128704/b128704.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(A128703(n)).

%e A128703(20) = 275 and there are 4 groups of order 275 (A000001(275) = 4), hence a(20) = 4.

%o (Magma) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n): n in [ h: h in [1..2000] | #t eq 2 and ((t[1, 1] lt 5 and t[1, 2] eq 1 and t[2, 1] eq 5 and t[2, 2] le 5) or (t[1, 1] eq 5 and t[1, 2] le 5 and t[2, 2] eq 1)) where t is Factorization(h) ] ];

%Y Cf. A000001 (number of groups of order n), A128703 (numbers of form 5^k*p, 1<=k<=5, p!=5 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, p>2 prime), A128694 (number of groups for orders of form 3^k*p, 1<=k<=6, p!=3 prime).

%K nonn

%O 1,1

%A _Klaus Brockhaus_, Mar 26 2007