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

1,1

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.

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 *)

Prog

(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

Crossrefs

Cf. A064899, A056868, A247371.

Sequence in context: A135711 A161543 A056868 * A060702 A054741 A177712

Adjacent sequences: A069206 A069207 A069208 * A069210 A069211 A069212

Keyword

nonn

Author

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

Extensions

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

Status

approved