A069209 Orders of non-Abelian Z-groups.

6, 10, 12, 14, 18, 20, 21, 22, 24, 26, 28, 30, 34, 36, 38, 39, 40, 42, 44, 46, 48, 50, 52, 54, 55, 56, 57, 58, 60, 62, 63, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 93, 94, 96, 98, 100, 102, 104, 105, 106, 108, 110, 111, 112, 114, 116, 117, 118

Offset: 1

Author

Sharon Sela (sharonsela(AT)hotmail.com), Apr 14 2002

Keywords

• nonn

Comments

Z-groups are groups in which all Sylow subgroups are cyclic. n belongs to this sequence iff n is divisible by two distinct primes p and q, such that p divides q-1. This sequence contains sequence A064899 and it is a subsequence of sequence A056868.

Numbers n such that there is more than one Z-group of order n. - Eric M. Schmidt, Sep 15 2014

Links

• Eric M. Schmidt, Table of n, a(n) for n = 1..10000
• Wikipedia, Z-group.

Crossrefs

Cf. A064899, A056868, A247371.

Programs

Maple

filter:= proc(n) local F,p,q; F:= numtheory:-factorset(n);
  for p in F do if member(1,map(`modp`,F,p)) then return true fi od:
  false
end proc:
select(filter, [$1..1000]); # Robert Israel, Sep 15 2014

Mathematica

filterQ[n_] := With[{pp = FactorInteger[n][[All, 1]]}, AnyTrue[pp, MemberQ[pp, q_ /; Divisible[q - 1, #]]&]];
Select[Range[2, 200], filterQ] (* Jean-François Alcover, Jul 18 2020 *)

Sage

def is_A069209(n) : return any((q-1)%p==0 for p,q in Combinations(prime_divisors(n),2)) # Eric M. Schmidt, Sep 15 2014

Extensions

Edited and extended by, and missing term 78 added by, Eric M. Schmidt, Sep 15 2014