A128706 Number of groups of order A128705(n).

2, 2, 1, 1, 1, 5, 1, 1, 6, 1, 2, 1, 2, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 15, 1, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 19, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 4, 2, 2, 1, 1, 1, 1, 2, 1, 5, 1, 1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 1

Offset

1,1

Comments

Number of groups for orders of form 7^k*p, where 1 <= k <= 4 and p is a prime different from 7.

The groups of these orders (up to A128705(64633879) = 7516192523 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(A128705(n)).

Example

A128705(30) = 686 and there are 15 groups of order 686 (A000001(686) = 15), hence a(30) = 15.

Prog

(Magma) D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n): n in [ h: h in [1..3500] | #t eq 2 and ((t[1, 1] lt 7 and t[1, 2] eq 1 and t[2, 1] eq 7 and t[2, 2] le 4) or (t[1, 1] eq 7 and t[1, 2] le 4 and t[2, 2] eq 1)) where t is Factorization(h) ] ];

Crossrefs

Cf. A000001 (number of groups of order n), A128705 (numbers of form 7^k*p, 1<=k<=4, p!=7 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, A128704 (number of groups for orders of form 5^k*p, 1<=k<=5, p!=5 prime).

Sequence in context: A242618 A180264 A225200 * A253586 A347621 A318191

Adjacent sequences: A128703 A128704 A128705 * A128707 A128708 A128709

Keyword

nonn

Author

Klaus Brockhaus, Mar 26 2007

Status

approved