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A226512 a(n) = minimum of sum of orders of all elements of G, where G is a group of order n.

%I A226512 #28 Dec 09 2023 07:10:39
%S A226512 1,3,7,7,21,13,43,15,25,31,111,31,157,57,147,31,273,43,343,71,85,133,
%T A226512 507,67,121,183,79,157,813,177,931,63,777,307,903,111,1333,381,235,
%U A226512 163,1641,183,1807,377,525,553,2163,127,337,171,1911,287,2757,133,331,351,457,871,3423,211
%N A226512 a(n) = minimum of sum of orders of all elements of G, where G is a group of order n.
%H A226512 G. Nebe, <a href="/A226512/b226512.txt">Table of n, a(n) for n = 1..200</a>
%H A226512 Habib Amiri and S. M. Jafarian Amiri, <a href="http://dx.doi.org/10.1142/S0219498811004057">Sum of element orders on finite groups of the same order</a>, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)
%H A226512 Habib Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, <a href="http://dx.doi.org/10.1080/00927870802502530">Sums of element orders in finite groups</a> Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)
%H A226512 Y. Marefat et al., <a href="http://dx.doi.org/10.1142/S0219498813500266">On the sum of element orders of finite simple groups</a>, J. Algebra Applications, 12 (2013), #1350026.
%F A226512 If p is prime, a(p) = A057660(p).
%e A226512 For n=6 the group S_3 is optimal: it has one element of order 1, 3 of order 2, and 2 of order 3, for a total of a(6) = 13.
%o A226512 (Magma) // Program from G. Nebe, Jun 14 2013
%o A226512 SS:=[];
%o A226512 for i in [1..200] do f:=[];
%o A226512 for j in [1..NumberOfSmallGroups(i)] do p:=0; G:=SmallGroup(i,j);
%o A226512 for g in G do p+:= Order(g); end for; Append(~f,p); end for;
%o A226512 Append(~SS,Minimum(f)); end for;
%o A226512 SS;
%Y A226512 Cf. A057660.
%K A226512 nonn
%O A226512 1,2
%A A226512 _N. J. A. Sloane_, Jun 13 2013
%E A226512 More terms from G. Nebe, Jun 14 2013