Search: id:A137315
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%I A137315
%S A137315 1,3,7,7,13,13,19,19,31,31,31,31,43,43,43,43,61,61,61,61,67,67,67,67,91,
%T A137315 91,91,91,91,91,91,91,121,121,121,121,127,127,127,127,151,151,151,151,
%U A137315 151,151,151,151
%N A137315 a(n) = least number m such that any finite group of order at least m 
               has at least n automorphisms.
%C A137315 a(n) <= (n-1)^(n + (n-2)[log_2(n-1)]) for n > 4 [Ledermann, Neumann, 
               Thm. 6.6].
%C A137315 a(n) is odd [MacHale, Sheehy, Thm. 15].
%C A137315 a(2n-1) = a(2n) for 1 < n < 204 [ibid.].
%C A137315 The case of cyclic groups shows that a(n)>=A139795(n). This inequality 
               can be strict: if M denotes the Mathieu group M_{22} of order 2^7.3^2.5.7.11, 
               then Aut(12.M) = M.2, so that a(2^8.3^2.5.7.11 + 1) > 2^9.3^3.5.7.11, 
               but A139795(2^8.3^2.5.7.11 + 1) = 2.3.5.7^2.11.13.23 + 1 < 2^9.3^3.5.7.11.
%D A137315 John N. Bray and Robert A. Wilson, On the orders of automorphism groups 
               of finite groups, Bull. London Math. Soc. 37 (2005) 381--385.
%D A137315 W. Ledermann, B. H. Neumann, On the order of the automorphism group of 
               a finite group 1, Proc. Roy. Soc. Lon., 233A(1195) (1956), 494-506
%D A137315 D. MacHale, R. Sheehy, Finite groups with few automorphisms, Math. Proc. 
               Roy. Irish Acad., 104A(2) (2004), 231-238
%H A137315 B. Jubin, <a href="http://math.berkeley.edu/~jubin/oeis.html">Sequences 
               contributed to the OEIS</a>.
%e A137315 a(3) = a(4) = 7 because every finite group with at least 7 elements has 
               at least 4 automorphisms while the cyclic group of order 6 has only 
               phi(6)=2 automorphisms.
%Y A137315 Different from A139795 (see Comments).
%Y A137315 Sequence in context: A109386 A024612 A073881 this_sequence A139795 A118259 
               A060845
%Y A137315 Adjacent sequences: A137312 A137313 A137314 this_sequence A137316 A137317 
               A137318
%K A137315 hard,more,nonn
%O A137315 1,2
%A A137315 Benoit Jubin (benoit_jubin(AT)yahoo.fr), Apr 06 2008, May 26 2008

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