A128694 Number of groups of order A128693(n).

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, 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, 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

Offset: 1

Author

Klaus Brockhaus, Mar 26 2007

Keywords

• nonn

Comments

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

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.

Examples

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

Links

• Klaus Brockhaus, Table of n, a(n) for n=1..10000
• Magma Computational Algebra System, Documentation, see Database of Small Groups.

Crossrefs

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).

Programs

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) ] ];

Formula

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