A298739 - OEIS

%I #23 Mar 22 2024 09:17:59

%S 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,

%T -1,-3,3,-3,50,-50,1,-1,13,-13,1,0,12,-13,5,-5,3,-2,0,-1,51,-50,3,-4,

%U 4,-4,14,-13,11,-11,0,-1,12,-12,1,2,263,-266,3,-3

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

%H Muniru A Asiru, <a href="/A298739/b298739.txt">Table of n, a(n) for n = 1..2046</a> [a(1023) and a(1024) corrected by Andrey Zabolotskiy]

%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="https://doi.org/10.1142/S0218196702001115">A Millennium Project: Constructing Small Groups</a>, Internat. J. Algebra and Computation, 12 (2002), 623-644.

%H Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubcay/">Numbers of Small Groups</a> [dead link]

%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>

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

%e There is only one group of order 1 and of order 2, so a(1) = A000001(2) - A000001(1) = 1 - 1 = 0.

%e There are 2 groups of order 4 and 3 is a cyclic number, so a(3) = A000001(4) - A000001(3) = 2 - 1 = 1.

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

%t (* Please note that, as of version 14, the Mathematica function FiniteGroupCount returns a wrong value for n = 1024 (49487365422 instead of 49487367289). *)

%t Differences[FiniteGroupCount[Range[100]]] (* _Paolo Xausa_, Mar 22 2024 *)

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

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

%K sign

%O 1,7

%A _Muniru A Asiru_, Jan 25 2018