A298739 First differences of A000001 (the number of groups of order n).

0, 0, 1, -1, 1, -1, 4, -3, 0, -1, 4, -4, 1, -1, 13, -13, 4, -4, 4, -3, 0, -1, 14, -13, 0, 3, -1, -3, 3, -3, 50, -50, 1, -1, 13, -13, 1, 0, 12, -13, 5, -5, 3, -2, 0, -1, 51, -50, 3, -4, 4, -4, 14, -13, 11, -11, 0, -1, 12, -12, 1, 2, 263, -266, 3, -3

Offset: 1

Author

Muniru A Asiru, Jan 25 2018

Keywords

• sign

Examples

			There is only one group of order 1 and of order 2, so a(1) = A000001(2) - A000001(1) = 1 - 1 = 0.
There are 2 groups of order 4 and 3 is a cyclic number, so a(3) = A000001(4) - A000001(3) = 2 - 1 = 1.
		

Links

• Muniru A Asiru, Table of n, a(n) for n = 1..2046 [a(1023) and a(1024) corrected by Andrey Zabolotskiy]
• H. U. Besche, B. Eick and E. A. O'Brien, A Millennium Project: Constructing Small Groups, Internat. J. Algebra and Computation, 12 (2002), 623-644.
• Gordon Royle, Numbers of Small Groups [dead link]
• Index entries for sequences related to groups

Crossrefs

Cf. A000001 (Number of groups of order n).

Programs

GAP

List([1..700],n -> NumberSmallGroups(n+1) - NumberSmallGroups(n));

Maple

with(GroupTheory): seq((NumGroups(n+1) - NumGroups(n), n=1..500));

Mathematica

(* Please note that, as of version 14, the Mathematica function FiniteGroupCount returns a wrong value for n = 1024 (49487365422 instead of 49487367289). *)
Differences[FiniteGroupCount[Range[100]]] (* Paolo Xausa, Mar 22 2024 *)

Formula

a(n) = A000001(n+1) - A000001(n).