cp's OEIS Frontend

A069209 Orders of non-Abelian Z-groups.

%I A069209 #23 Jul 18 2020 09:01:49
%S A069209 6,10,12,14,18,20,21,22,24,26,28,30,34,36,38,39,40,42,44,46,48,50,52,
%T A069209 54,55,56,57,58,60,62,63,66,68,70,72,74,76,78,80,82,84,86,88,90,92,93,
%U A069209 94,96,98,100,102,104,105,106,108,110,111,112,114,116,117,118
%N A069209 Orders of non-Abelian Z-groups.
%C A069209 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.
%C A069209 Numbers n such that there is more than one Z-group of order n. - _Eric M. Schmidt_, Sep 15 2014
%H A069209 Eric M. Schmidt, <a href="/A069209/b069209.txt">Table of n, a(n) for n = 1..10000</a>
%H A069209 Wikipedia, <a href="http://en.wikipedia.org/wiki/Z-group">Z-group</a>.
%p A069209 filter:= proc(n) local F,p,q; F:= numtheory:-factorset(n);
%p A069209   for p in F do if member(1,map(`modp`,F,p)) then return true fi od:
%p A069209   false
%p A069209 end proc:
%p A069209 select(filter, [$1..1000]); # _Robert Israel_, Sep 15 2014
%t A069209 filterQ[n_] := With[{pp = FactorInteger[n][[All, 1]]}, AnyTrue[pp, MemberQ[pp, q_ /; Divisible[q - 1, #]]&]];
%t A069209 Select[Range[2, 200], filterQ] (* _Jean-François Alcover_, Jul 18 2020 *)
%o A069209 (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
%Y A069209 Cf. A064899, A056868, A247371.
%K A069209 nonn
%O A069209 1,1
%A A069209 Sharon Sela (sharonsela(AT)hotmail.com), Apr 14 2002
%E A069209 Edited and extended by, and missing term 78 added by, _Eric M. Schmidt_, Sep 15 2014