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.

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

Author

R. J. Mathar, Jan 07 2024

Keywords

• nonn
• easy

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.

Crossrefs

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

Programs

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: