  In [Akbari and Moghaddamfar, Recognizing by order and degree pattern of some
projective special linear groups, {\it Internat. J. Algebra Comput.}, 2012] the
authors possed the following problem: \\ {\bf Problem.} {\it Is there a simple
group which is $k$-fold OD-characterizable for $k\geq3\ ?$ }
  In this paper as the main result we give positive answer to the above problem
and we introduce two simple groups which are $k$-fold OD-characterizable such
that $k\geq6$.
  Also in [R. Kogani-Moghadam and A. R. Moghaddamfar, Groups with the same
order and degree pattern, {\it Science China Mathematics}, 2012], the authors
possed two conjectures as follows: \\ {\bf Conjecture 1.} {\it All alternating
groups $A_m$ with $m \not= 10$ are OD-characterizable.} \\ {\bf Conjecture 2.}
{\it All symmetric groups $S_m$, with $m \not= 10$, are $n$-fold
OD-characterizable, where $n\in\{1, 3\}$.}
  In this paper we find some alternating and some symmetric groups such that
these conjectures are not true for them.
