A279229 Odd orders n for which a complete dihedral Hamiltonian cycle system of the cocktail graph exists.

21, 33, 45, 57, 65, 69, 77, 85, 93, 105, 117, 123, 129, 133, 141, 145, 153, 161, 165, 177, 185, 189, 201, 209, 213, 217, 219, 221, 225, 237, 245, 249, 253, 261, 265, 267, 273, 285, 287, 291, 297, 301, 305, 309, 321, 325, 329, 333, 341, 345, 357

Offset: 1

Author

R. J. Mathar, Jan 04 2017

Keywords

• nonn
• easy

Links

• M. Buratti and F. Merola, Dihedral Hamiltonian cycle systems of the Cocktail Party Graph, J. Combin. Des. 21 (1) (2013) 1-23, Section 3.

Programs

Maple

isA000961 := proc(n)
    local pf;
    if n = 1 then
        return true;
    end if;
    pf := ifactors(n)[2] ;
    if nops(pf) > 1 then
        false;
    else
        true;
    end if ;
end proc:
A023506 := proc(p)
    padic[ordp](p-1,2) ;
end proc:
isA279229 := proc(n)
    local ct2,p,l ;
    if type(n,'even') then
        false;
    elif isA000961(n) then
        false;
    else
        ct2 := 0 ;
        for pf in ifactors(n)[2] do
            l := A023506(op(1,pf)) ;
            ct2 := ct2+l*op(2,pf) ;
        end do:
        type(ct2,'even') ;
    end if;
end proc:
for n from 2 to 2000 do
    if isA279229(n) then
        printf("%d,",n);
    end if;
end do:

Mathematica

A023506[p_] := IntegerExponent[p - 1, 2];
isA279229[n_] := Module[{ct2, l}, Which[EvenQ[n], False, PrimePowerQ[n], False, True, ct2 = 0; Do[l = A023506[pf[[1]]]; ct2 = ct2 + l*pf[[2]], {pf, FactorInteger[n]}]; EvenQ[ct2]]];
Select[Range[2, 400], isA279229] (* Jean-François Alcover, Oct 28 2023, after R. J. Mathar's program *)