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Numbers n such that the number of groups of order n exceeds the sum of divisors of n.
1

%I #22 Feb 19 2019 10:05:55

%S 64,128,192,256,288,320,384,448,512,576,640,768,864,896,960,1024,1152,

%T 1280,1344,1408,1440,1536,1600,1664,1728,1792,1920,2048,2112,2176,2187

%N Numbers n such that the number of groups of order n exceeds the sum of divisors of n.

%C It seems that 1 is the only number such that the number of groups equals the sum of the divisors and that for almost all numbers the sum of the divisors exceeds the number of groups.

%H H. U. Besche, B. Eick and E. A. O'Brien, <a href="http://dx.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>, June 2000.

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

%F Sequence is { n | A000001(n) > A000203(n) }.

%e 64 is in the sequence because 267 = A000001(64) > A000203(64) = 127.

%e 128 is in the sequence because 2328 = A000001(128) > A000203(128) = 255.

%e 1920 is in the sequence because 241004 = A000001(1920) > A000203(1920) = 6120.

%p with(GroupTheory): with(numtheory):

%p for n from 1 to 2047 do if NumGroups(n) > sigma(n) then print(n); fi; od;

%t Select[Range[10^4], FiniteGroupCount[#] > DivisorSigma[1, #] &] (* _Amiram Eldar_, Feb 19 2019 *)

%o (GAP) A296166 := Filtered([1..2015], n -> NumberSmallGroups(n) > Sigma(n));

%Y Subsequence of A090052.

%Y Cf. A000001, A000203.

%K nonn,more

%O 1,1

%A _Muniru A Asiru_, Dec 06 2017