

A060702


Orders of finite groups that have trivial center.


2



1, 6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 30, 34, 36, 38, 39, 42, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 66, 68, 70, 72, 74, 75, 78, 80, 82, 84, 86, 90, 93, 94, 96, 98, 100, 102, 106, 108, 110, 111, 114, 116, 118, 120, 122, 126, 129, 130, 132, 134, 136
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OFFSET

1,2


COMMENTS

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.
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 q1.  Des MacHale and Mossie Crowe, Jul 05 2005
This sequence is closed under multiplication.  Eric M. Schmidt, Aug 27 2012


LINKS

Eric M. Schmidt, Table of n, a(n) for n = 1..1000


EXAMPLE

The symmetric group S_3 of order 6 has trivial center so 6 belongs to the sequence.


CROSSREFS

Cf. A056868, A059806.
Sequence in context: A161543 A056868 A069209 * A054741 A177712 A176552
Adjacent sequences: A060699 A060700 A060701 * A060703 A060704 A060705


KEYWORD

nonn


AUTHOR

Ahmed Fares (ahmedfares(AT)mydeja.com), Apr 20 2001


EXTENSIONS

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
Sequence extended and corrected by Eric M. Schmidt, Aug 27 2012


STATUS

approved



