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

1, 3, 7, 7, 21, 13, 43, 15, 25, 31, 111, 31, 157, 57, 147, 31, 273, 43, 343, 71, 85, 133, 507, 67, 121, 183, 79, 157, 813, 177, 931, 63, 777, 307, 903, 111, 1333, 381, 235, 163, 1641, 183, 1807, 377, 525, 553, 2163, 127, 337, 171, 1911, 287, 2757, 133, 331, 351, 457, 871, 3423, 211

Offset: 1

Author

N. J. A. Sloane, Jun 13 2013

Keywords

• nonn

Examples

			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.
		

Links

• G. Nebe, Table of n, a(n) for n = 1..200
• Habib Amiri and S. M. Jafarian Amiri, Sum of element orders on finite groups of the same order, J. Algebra Appl. 10 (2011), no. 2, 187--190. MR2795731 (2012d:20050)
• Habib Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, Sums of element orders in finite groups Comm. Algebra 37 (2009), no. 9, 2978--2980. MR2554185 (2010i:20022)
• Y. Marefat et al., On the sum of element orders of finite simple groups, J. Algebra Applications, 12 (2013), #1350026.

Crossrefs

Cf. A057660.

Programs

Magma

// Program from G. Nebe, Jun 14 2013
SS:=[];
for i in [1..200] do f:=[];
for j in [1..NumberOfSmallGroups(i)] do p:=0; G:=SmallGroup(i,j);
for g in G do p+:= Order(g); end for; Append(~f,p); end for;
Append(~SS,Minimum(f)); end for;
SS;

Formula

If p is prime, a(p) = A057660(p).

Extensions

More terms from G. Nebe, Jun 14 2013