A098885 -id:A098885 - cp's OEIS frontend

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

Klaus Brockhaus, Mar 13 2007

nonn

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)

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

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

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).

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)) } ] ];

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

A384607 Possible values for the number of groups of order equal to a prime power, in order of size.

Original entry on oeis.org

1, 2, 5, 14, 15, 51, 67, 77, 83, 87, 97, 101, 107, 111, 125, 131, 145, 149, 155, 159, 173, 183, 193, 203, 207, 217, 227, 231, 245, 265, 267, 269, 275, 279, 289, 293, 323, 327, 341, 347, 365, 371, 385, 395, 399, 413, 423, 433, 447, 457, 461, 467, 491, 504, 515

Offset: 1

Robin Jones, Jun 04 2025

nonn

Possible values of A098885, ordered by size.

This sequence is the same regardless of whether 1 is considered a prime power or not (see A000961 for discussion on this) as A000001(1)=A000001(p)=1 for all p.

			1 is in this sequence because A000001(2) = 1.
2 is in this sequence because A000001(2^2) = 2.
5 is in this sequence because A000001(2^3) = 5.
3 is not in this sequence as no prime power p^k has A000001(p^k)=3.

Robin Jones, Table of n, a(n) for n = 1..180

Cf. A000001, A098885, A384606.

A098886 Number of nonisomorphic groups with prime power order p^m, m>1.

Original entry on oeis.org

1, 2, 5, 2, 14, 2, 5, 51, 2, 267, 15, 2, 5, 2328, 2, 67, 56092, 2, 5, 2, 10494213, 2, 15, 504, 2, 2, 49487367289, 5, 2, 2, 2

Offset: 1

Lekraj Beedassy, Oct 14 2004

nonn

H. U. Besche, B. Eick and E. A. O'Brien, Number of isomorphism types of finite groups of given order
David Burrell, On the number of groups of order 1024, Communications in Algebra, 2021, 1-3.

Cf. A000001, A025475, A098885, A128604.

/* Program returns -1 for an order o if the groups of that order are not contained in the Small Groups Library */ D := SmallGroupDatabase(); S := []; for o in [1..2047] do if (o eq 1 or IsPrimePower(o)) and not IsPrime(o) then if IsInSmallGroupDatabase(D, o) then a := NumberOfSmallGroups(D, o); else a := -1; end if; Append(~S, a); end if; end for; S; /* Klaus Brockhaus, Mar 15 2007 */

a(n) = A000001(A025475(n)).

a(30) corrected by Klaus Brockhaus, Mar 15 2007

a(27) corrected by David Burrell, Jun 06 2022

A384606 Possible values for the number of groups of order equal to a prime power, in order of first appearance.

Original entry on oeis.org

1, 2, 5, 14, 51, 267, 15, 2328, 67, 56092, 10494213, 504, 49487367289

Offset: 1

Robin Jones, Jun 04 2025

Equal A098885 with the duplicate entries removed.
a(14) = A000001(2048) (this value is currently unknown).
This sequence is the same regardless of whether 1 is considered a prime power or not (see A000961 for discussion on this) as A000001(1) = A000001(p) = 1 for all p.

			1 is in this sequence because A000001(2) = 1.
2 is in this sequence because A000001(2^2) = 2.
5 is in this sequence because A000001(2^3) = 5.
3 is not in this sequence as no prime power p^k has A000001(p^k)=3.

Cf. A098885, A384607.

cp's OEIS Frontend

A128604 Number of groups of order A128603(n).

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A384607 Possible values for the number of groups of order equal to a prime power, in order of size.

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A098886 Number of nonisomorphic groups with prime power order p^m, m>1.

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A384606 Possible values for the number of groups of order equal to a prime power, in order of first appearance.

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