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A368832
Integers not of one of the 5 forms p^k, p*q^k, 2*p*q^k, p*q*r or 2*p*q*r with p, q, r distinct primes and k>=0.
0
36, 60, 72, 84, 100, 108, 120, 132, 140, 144, 156, 168, 180, 196, 200, 204, 216, 220, 225, 228, 240, 252, 260, 264, 276, 280, 288, 300, 308, 312, 315, 324, 336, 340, 348, 360, 364, 372, 380, 392, 396, 400, 408, 420, 432, 440, 441, 444, 450, 456, 460, 468, 476, 480, 484, 492, 495, 500, 504, 516, 520, 525, 528, 532, 540, 552
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
Cf. A051270 (subsequence), A036785 (subsequence), A074969 (subsequence).
Sequence in context: A260138 A260131 A320632 * A188633 A328961 A335295
KEYWORD
nonn,easy
AUTHOR
R. J. Mathar, Jan 07 2024
STATUS
approved