A337705 - OEIS

%I #15 Oct 24 2020 16:52:17

%S 1,3,7,11,13,15,19,21,23,25,27,31,33,39,43,45,47,49,55,57,59,61,63,67,

%T 71,73,75,77,79,83,85,87,95,97,99,101,103,105,111,113,115,119,121,125,

%U 127,129,133,135,143,147,151,153,157,159,161,163

%N Possible sums of orders of elements of finite groups.

%C The sum of the orders of all elements of a finite group G is denoted by psi(G).

%C psi(A X B) = psi(A)*psi(B) for finite groups A and B of coprime orders.

%C psi(G) <= 7/11 psi(C_n) < psi(C_n) for every noncyclic finite group G of order n.

%C psi(G) < 1/(p - 1) psi(C_n) for every noncyclic finite group G of order n, where p the smallest prime divisor of n.

%C Conjecture: If S is a simple group and G is a soluble group satisfying |S|=|G|, then psi(S) < psi(G).

%H M. Farrokhi D. G., <a href="/A337705/b337705.txt">Table of n, a(n) for n = 1..1027</a>

%H H. Amiri, S. M. Jafarian Amiri, and I. M. Isaacs, <a href="https://doi.org/10.1080/00927870802502530">Sums of element orders in finite groups</a>, Comm. Algebra 37(9) (2009), 2978-2980.

%H M. Herzog, P. Longobardi, and M. Maj, <a href="https://doi.org/10.1007/978-981-13-2047-7_4">Properties of finite and periodic groups determined by their element of orders (a survey)</a>, Group theory and computation, 59-90, Indian Stat. Inst. Ser., Springer, Singapore, 2018.

%e psi(C_6) = 1 + 2 + 3 + 3 + 6 + 6 = 21.

%o (GAP) Sum(List(G, Order));

%K nonn

%O 1,2

%A _M. Farrokhi D. G._, Sep 16 2020