The vortex coronagraph is one of the most promising coronagraphs for high-contrast imaging because of its simplicity, small inner working angle, high throughput, and clear off-axis discovery space. However, as with most coronagraphs, centrally obscured on-axis telescopes degrade contrast. Based on the remarkable ability of vortex coronagraphs to move light between the interior and exterior of pupils, we propose a method based on multiple vortices, that without sacrificing throughput, reduces the residual light leakage to (a/A)(n), with n ≥ 4, and a and A being the radii of the central obscuration and primary mirror, respectively. This method thus enables high contrasts to be reached even with an on-axis telescope.