A recent infinite decomposability result shows that, for any integer $m$, a random variable following the exponential distribution can be written as the sum of $m$ discontinuous random variables and another one following the exponential distribution, all of them independent. This note extends this result to the Gamma, Laplace and $n$-Laplace distributions, with a clear identification on the involved discontinuous distribution. We also discuss some properties of this new discontinuous distribution, proving that it is also infinitely decomposable.