The polarized longitudinal-transverse structure function $\sigma_{LT^\prime}$ has been measured using the $p(\vec e,e'\pi^+)n$ reaction in the $\Delta(1232)$ resonance region at $Q^2=0.40$ and 0.65 GeV$^2$. No previous $\sigma_{LT^\prime}$ data exist for this reaction channel. The kinematically complete experiment was performed at Jefferson Lab with the CEBAF Large Acceptance Spectrometer (CLAS) using longitudinally polarized electrons at an energy of 1.515 GeV. A partial wave analysis of the data shows generally better agreement with recent phenomenological models of pion electroproduction compared to the previously measured $\pi^0 p$ channel. A fit to both $\pi^0 p$ and $\pi^+ n$ channels using a unitary isobar model suggests the unitarized Born terms provide a consistent description of the non-resonant background. The $t$-channel pion pole term is important in the $\pi^0 p$ channel through a rescattering correction, which could be model-dependent.
