  Inspirations for this paper can be traced to Urbanik (1972) where convolution
semigroups of multiple decomposable distributions were introduced. In
particular, the classical gamma $\mathbb{G}_t$ and $\log \mathbb{G}_t$, $t>0$
variables are selfdecomposable. In fact, we show that $\log \mathbb{G}_t$ is
twice selfdecomposable if, and only if, $t\geq t_1 \approx 0.15165$. Moreover,
we provide several new factorizations of the Gamma function and the Gamma
distributions. To this end, we revisit the class of multiply selfdecomposable
distributions, denoted $L_n(R)$, and propose handy tools for its
characterization, mainly based on the Mellin-Euler's differential operator.
Furthermore, we also give a perspective of generalization of the class $L_n(R)$
based on linear operators or on stochastic integral representations.
