%I #20 Oct 20 2022 07:43:12
%S 1,6,10,12,14,18,20,21,22,24,26,30,34,36,38,39,42,46,48,50,52,54,55,
%T 56,57,58,60,62,66,68,70,72,74,75,78,80,82,84,86,90,93,94,96,98,100,
%U 102,106,108,110,111,114,116,118,120,122,126,129,130,132,134,136
%N Orders of finite groups that have trivial center.
%C Apart from the first element 1 this is a subsequence of A056868 because a nilpotent group has nontrivial center. If n = 0 mod 6 or n >= 6 and n = 2 mod 4 then n is in this sequence.
%C If n >= 6 and n == 2 mod 4 then n is a member of the sequence because of the dihedral group of order 2(2k+1). In addition, if p is a prime and p == 1 mod 4 then n=4p is a member of the sequence; however, if p == 3 mod 4 and p>5, then n=4p is not a member of the sequence. Furthermore, if n=pq where p and q are distinct odd primes with p<q, then pq belongs to the sequence if and only if p divides q-1. - _Des MacHale_ and Mossie Crowe, Jul 05 2005
%C This sequence is closed under multiplication. - _Eric M. Schmidt_, Aug 27 2012
%H Eric M. Schmidt, <a href="/A060702/b060702.txt">Table of n, a(n) for n = 1..1000</a>
%e The symmetric group S_3 of order 6 has trivial center so 6 belongs to the sequence.
%Y Cf. A056868, A059806.
%Y For the corresponding numbers of centerless groups of these orders see A357900.
%K nonn
%O 1,2
%A Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 20 2001
%E The old entry 89 was an error, since it is a prime. - Robert F. Bailey (robertb(AT)math.carleton.ca) and Brett Stevens (brett(AT)math.carleton.ca), Jul 16 2009
%E Sequence extended and corrected by _Eric M. Schmidt_, Aug 27 2012
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