Intercalant and intermolecular phonon assisted superconductivity in K-doped picene

K$_3$-picene is a superconducting molecular crystal with critical temperature T$_c=7$K or 18K, depending on preparation conditions. Using density functional theory we show that electron-phonon interaction accounts for T$_c ~ 3-8$ K. The average electron-phonon coupling, calculated by including the phonon energy scale in the electron-phonon scattering, is $\lambda=0.73$ and $\omega_{\rm log}=18.0\,$ meV. Intercalant and intermolecular phonon-modes contribute substantially (40%) to $\lambda$ as also shown by the isotope exponents of potassium (0.19) and carbon (0.31). The relevance of these modes makes superconductivity in K-doped picene peculiar and different from that of fullerenes.