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

%I A298739 #23 Mar 22 2024 09:17:59
%S A298739 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 A298739 -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 A298739 4,-4,14,-13,11,-11,0,-1,12,-12,1,2,263,-266,3,-3
%N A298739 First differences of A000001 (the number of groups of order n).
%H A298739 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 A298739 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 A298739 Gordon Royle, <a href="http://staffhome.ecm.uwa.edu.au/~00013890/remote/cubcay/">Numbers of Small Groups</a> [dead link]
%H A298739 <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%F A298739 a(n) = A000001(n+1) - A000001(n).
%e A298739 There is only one group of order 1 and of order 2, so a(1) = A000001(2) - A000001(1) = 1 - 1 = 0.
%e A298739 There are 2 groups of order 4 and 3 is a cyclic number, so a(3) = A000001(4) - A000001(3) = 2 - 1 = 1.
%p A298739 with(GroupTheory): seq((NumGroups(n+1) - NumGroups(n), n=1..500));
%t A298739 (* Please note that, as of version 14, the Mathematica function FiniteGroupCount returns a wrong value for n = 1024 (49487365422 instead of 49487367289). *)
%t A298739 Differences[FiniteGroupCount[Range[100]]] (* _Paolo Xausa_, Mar 22 2024 *)
%o A298739 (GAP) List([1..700],n -> NumberSmallGroups(n+1) - NumberSmallGroups(n));
%Y A298739 Cf. A000001 (Number of groups of order n).
%K A298739 sign
%O A298739 1,7
%A A298739 _Muniru A Asiru_, Jan 25 2018