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

1,1

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.

Links

Klaus Brockhaus, Table of n, a(n) for n=1..10000

MAGMA Documentation, Database of Small Groups

Formula

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

Example

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

Prog

(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) ] ];

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).

Sequence in context: A319171 A047888 A330964 * A075259 A307877 A259703

Adjacent sequences: A128701 A128702 A128703 * A128705 A128706 A128707

Keyword

nonn

Author

Klaus Brockhaus, Mar 26 2007

Status

approved