A368832 - OEIS

%I #6 Jan 07 2024 09:44:47

%S 36,60,72,84,100,108,120,132,140,144,156,168,180,196,200,204,216,220,

%T 225,228,240,252,260,264,276,280,288,300,308,312,315,324,336,340,348,

%U 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 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 Cyclic groups of these orders cannot be Schur groups, see the Theorem by [Evdokimov et al.].

%H 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 isA007304 := proc(n)

%p     if bigomega(n) = 3 and A001221(n) =3 then

%p         true;

%p     else

%p         false ;

%p     end if;

%p end proc:

%p # list of prime exponents

%p pexp := proc(n)

%p     local e,pe ;

%p     e := [] ;

%p     for pe in ifactors(n)[2] do

%p         e := [op(e),op(2,pe)] ;

%p     end do:

%p     e ;

%p end proc:

%p isCycSchGr := proc(n)

%p     local om,nhalf ,pe;

%p     om := A001221(n) ;

%p     if  om > 4 then

%p         return false;

%p     elif om = 4 then

%p         # require 2*p*q*r

%p         if type(n,'even') and type(n/2,'odd') then

%p             nhalf := n/2 ;

%p             # require nhalf =p*q*r in A007304

%p             return isA007304(nhalf) ;

%p         else

%p             false;

%p         end if;

%p     elif om = 3 then

%p         # require p*q*r or 2*p*q^k

%p         if type(n,'even') and type(n/2,'odd') then

%p             nhalf := n/2 ;

%p             # require nhalf =p*q^k

%p             pe := pexp(nhalf) ;

%p             if nops(pe) =2 and 1 in convert(pe,set) then

%p                 true;

%p             else

%p                 false ;

%p             end if;

%p         elif type(n,'odd') then

%p             # require n =p*q*r

%p             if isA007304(n) then

%p                 true;

%p             else

%p                 false ;

%p             end if;

%p         else

%p             false;

%p         end if;

%p     elif om = 2 then

%p         # require p*q^k

%p         pe := pexp(n) ;

%p         if 1 in convert(pe,set) then

%p             true;

%p         else

%p             false;

%p         end if;

%p     else

%p         # p^k, k>=0

%p         true ;

%p     end if;

%p end proc:

%p for n from 1 to 3000 do

%p     if not isCycSchGr(n) then

%p         printf("%d,",n) ;

%p     end if;

%p end do:

%Y Cf. A051270 (subsequence), A036785 (subsequence), A074969 (subsequence).

%K nonn,easy

%O 1,1

%A _R. J. Mathar_, Jan 07 2024