How many ideals whose quotient rings are Gorenstein exist?

For an Ulrich ideal in a Gorenstein local ring, the quotient ring is again Gorenstein. Aiming to further develop the theory of Ulrich ideals, this paper investigates a naive question of how many non-principal ideals whose quotient rings are Gorenstein exist in a given Gorenstein ring. The main result provides that the number of such graded ideals in a symmetric numerical semigroup ring $R$ coincides with the conductor of the semigroup. We furthermore provide a complete list of non-principal graded ideals $I$ in $R$ whose quotient rings $R\mathrm{/}I$ are Gorenstein.