OFFSET
1,1
COMMENTS
Cyclic groups of these orders cannot be Schur groups, see the Theorem by [Evdokimov et al.].
LINKS
S. Evdokimov, I. Kovacs, and I. Ponomarenko, On Schurity of Finite Abelian Groups, Comm. Algebra 44 (2016) 101-117, see the Cyclic Schur Group Theorem.
MAPLE
isA007304 := proc(n)
if bigomega(n) = 3 and A001221(n) =3 then
true;
else
false ;
end if;
end proc:
# list of prime exponents
pexp := proc(n)
local e, pe ;
e := [] ;
for pe in ifactors(n)[2] do
e := [op(e), op(2, pe)] ;
end do:
e ;
end proc:
isCycSchGr := proc(n)
local om, nhalf , pe;
om := A001221(n) ;
if om > 4 then
return false;
elif om = 4 then
# require 2*p*q*r
if type(n, 'even') and type(n/2, 'odd') then
nhalf := n/2 ;
# require nhalf =p*q*r in A007304
return isA007304(nhalf) ;
else
false;
end if;
elif om = 3 then
# require p*q*r or 2*p*q^k
if type(n, 'even') and type(n/2, 'odd') then
nhalf := n/2 ;
# require nhalf =p*q^k
pe := pexp(nhalf) ;
if nops(pe) =2 and 1 in convert(pe, set) then
true;
else
false ;
end if;
elif type(n, 'odd') then
# require n =p*q*r
if isA007304(n) then
true;
else
false ;
end if;
else
false;
end if;
elif om = 2 then
# require p*q^k
pe := pexp(n) ;
if 1 in convert(pe, set) then
true;
else
false;
end if;
else
# p^k, k>=0
true ;
end if;
end proc:
for n from 1 to 3000 do
if not isCycSchGr(n) then
printf("%d, ", n) ;
end if;
end do:
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Jan 07 2024
STATUS
approved