A128704 Number of groups of order A128703(n).

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, 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, 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

Offset: 1

Author

Klaus Brockhaus, Mar 26 2007

Keywords

• nonn

Comments

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

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.

Examples

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

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), 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).

Programs

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) ] ];

Formula

a(n) = A000001(A128703(n)).