A279229 - OEIS

%I #45 Oct 28 2023 05:45:18

%S 21,33,45,57,65,69,77,85,93,105,117,123,129,133,141,145,153,161,165,

%T 177,185,189,201,209,213,217,219,221,225,237,245,249,253,261,265,267,

%U 273,285,287,291,297,301,305,309,321,325,329,333,341,345,357

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

%H M. Buratti and F. Merola, <a href="http://dx.doi.org/10.1002/jcd.21311">Dihedral Hamiltonian cycle systems of the Cocktail Party Graph</a>, J. Combin. Des. 21 (1) (2013) 1-23, Section 3.

%p isA000961 := proc(n)

%p     local pf;

%p     if n = 1 then

%p         return true;

%p     end if;

%p     pf := ifactors(n)[2] ;

%p     if nops(pf) > 1 then

%p         false;

%p     else

%p         true;

%p     end if ;

%p end proc:

%p A023506 := proc(p)

%p     padic[ordp](p-1,2) ;

%p end proc:

%p isA279229 := proc(n)

%p     local ct2,p,l ;

%p     if type(n,'even') then

%p         false;

%p     elif isA000961(n) then

%p         false;

%p     else

%p         ct2 := 0 ;

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

%p             l := A023506(op(1,pf)) ;

%p             ct2 := ct2+l*op(2,pf) ;

%p         end do:

%p         type(ct2,'even') ;

%p     end if;

%p end proc:

%p for n from 2 to 2000 do

%p     if isA279229(n) then

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

%p     end if;

%p end do:

%t A023506[p_] := IntegerExponent[p - 1, 2];

%t 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]]];

%t Select[Range[2, 400], isA279229] (* _Jean-François Alcover_, Oct 28 2023, after _R. J. Mathar_'s program *)

%K nonn,easy

%O 1,1

%A _R. J. Mathar_, Jan 04 2017