  We investigate the transverse momentum weighted azimuthal asymmetry with a
$\sin\left(3\phi_h -\phi_S\right)$ modulation in electroproduction of charged
pion production off the transversely polarized proton. The asymmetry is
contributed by the product of the second transverse-moment of the pretzelosity
distribution function and the first transverse-moment of the Collins function.
Using the extracted result for the proton pretzelosity and the available
parametrization for the pion Collins function, we numerically predict the
$P_{hT}^3$-weighted $\sin\left(3\phi_h -\phi_S\right)$ asymmetry for charged
pion production at electron ion colliders (EICs). It is found that the weighted
asymmetry is around 2 percent in the kinematics region of EICs. Our results
show that the planned high luminosity electron ion colliders could be feasible
to access pretzelosity in the near future.
