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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.

%I A368832 #6 Jan 07 2024 09:44:47
%S A368832 36,60,72,84,100,108,120,132,140,144,156,168,180,196,200,204,216,220,
%T A368832 225,228,240,252,260,264,276,280,288,300,308,312,315,324,336,340,348,
%U A368832 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
%N 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.
%C A368832 Cyclic groups of these orders cannot be Schur groups, see the Theorem by [Evdokimov et al.].
%H A368832 S. Evdokimov, I. Kovacs, and I. Ponomarenko, <a href="https://doi.org/10.101080/00927872.2024.958848">On Schurity of Finite Abelian Groups</a>, Comm. Algebra 44 (2016) 101-117, see the Cyclic Schur Group Theorem.
%p A368832 isA007304 := proc(n)
%p A368832     if bigomega(n) = 3 and A001221(n) =3 then
%p A368832         true;
%p A368832     else
%p A368832         false ;
%p A368832     end if;
%p A368832 end proc:
%p A368832 # list of prime exponents
%p A368832 pexp := proc(n)
%p A368832     local e,pe ;
%p A368832     e := [] ;
%p A368832     for pe in ifactors(n)[2] do
%p A368832         e := [op(e),op(2,pe)] ;
%p A368832     end do:
%p A368832     e ;
%p A368832 end proc:
%p A368832 isCycSchGr := proc(n)
%p A368832     local om,nhalf ,pe;
%p A368832     om := A001221(n) ;
%p A368832     if  om > 4 then
%p A368832         return false;
%p A368832     elif om = 4 then
%p A368832         # require 2*p*q*r
%p A368832         if type(n,'even') and type(n/2,'odd') then
%p A368832             nhalf := n/2 ;
%p A368832             # require nhalf =p*q*r in A007304
%p A368832             return isA007304(nhalf) ;
%p A368832         else
%p A368832             false;
%p A368832         end if;
%p A368832     elif om = 3 then
%p A368832         # require p*q*r or 2*p*q^k
%p A368832         if type(n,'even') and type(n/2,'odd') then
%p A368832             nhalf := n/2 ;
%p A368832             # require nhalf =p*q^k
%p A368832             pe := pexp(nhalf) ;
%p A368832             if nops(pe) =2 and 1 in convert(pe,set) then
%p A368832                 true;
%p A368832             else
%p A368832                 false ;
%p A368832             end if;
%p A368832         elif type(n,'odd') then
%p A368832             # require n =p*q*r
%p A368832             if isA007304(n) then
%p A368832                 true;
%p A368832             else
%p A368832                 false ;
%p A368832             end if;
%p A368832         else
%p A368832             false;
%p A368832         end if;
%p A368832     elif om = 2 then
%p A368832         # require p*q^k
%p A368832         pe := pexp(n) ;
%p A368832         if 1 in convert(pe,set) then
%p A368832             true;
%p A368832         else
%p A368832             false;
%p A368832         end if;
%p A368832     else
%p A368832         # p^k, k>=0
%p A368832         true ;
%p A368832     end if;
%p A368832 end proc:
%p A368832 for n from 1 to 3000 do
%p A368832     if not isCycSchGr(n) then
%p A368832         printf("%d,",n) ;
%p A368832     end if;
%p A368832 end do:
%Y A368832 Cf. A051270 (subsequence), A036785 (subsequence), A074969 (subsequence).
%K A368832 nonn,easy
%O A368832 1,1
%A A368832 _R. J. Mathar_, Jan 07 2024