We discovered a specific property of the eigenmodes of a cylindrical waveguide, due to which efficient mode excitation at ultrahigh (s=4n+1, n is integer) multiples of the gyrofrequency can be provided by a polyhelical weakly relativistic electron beam, standardly used for gyrotron operation. In the proof-of-principle experiment with a V-band gyrotron driven by a 25-keV, 2-A beam, about 100 mW radiation power at the fifth cyclotron harmonic (0.22 THz) has been detected in the cw regime. Based on the gyrotron theory, we demonstrate in simulations that, using parameters of existing gyrotrons, this method can be applied for producing cw radiation at a power level of up to several watts in the terahertz gap.