A128604 Number of groups of order A128603(n).

1, 1, 2, 1, 1, 5, 2, 1, 1, 14, 1, 1, 1, 2, 5, 1, 1, 51, 1, 1, 1, 1, 2, 1, 1, 1, 267, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 2, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 67, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Klaus Brockhaus, Mar 13 2007

Keywords

• nonn

Comments

Number of groups whose order divides p^6 for p a prime.

The groups of these orders (up to A128603(54403784) = 1073741789 in version V2.13-4) form a class contained in the Small Groups Library of MAGMA. (corrected Mar 18 2007)

Examples

			A128603(10) = 16 and there are 14 groups of order 16 (A000001(16) = 14), hence a(10) = 14.
		

Links

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

Crossrefs

Cf. A000001 (number of groups of order n), A128603 (numbers dividing p^6 for p a prime), A098885 (number of groups of prime power orders).

Programs

Magma

D:=SmallGroupDatabase(); [ NumberOfSmallGroups(D, n) : n in [ k: k in [1..455] | exists(t) {x: x in [t: t in [1..6] ] | IsPower(k, x) and IsPrime(Iroot(k, x)) } ] ];

Formula

a(n) = A000001(A128603(n)).