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
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