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alternating n.  and  Friedman n.

First numbers that belong to both sets:

25, 121, 125, 127, 216, 343, 347, 1296, 2187, 2349, 2501, 2503, 2505, 2507, 2509, 4167, 5832, 6145, 9216, 10521, 10525, 12101, 12103, 12105, 12107, 12109, 12167, 12321, 12505, 12769, 14129, 16347, 16381, 16385, 16387, 16743, 16765, 16783, 18723, 21870, 23456, 23490, 25474, 25872, 27436, 27634, 27654, 29632, 32747, 32761, 32765, 32781, 32783, 32785, 36549, 41472, 43692, 45056, 45632, 47652, 49896, 52947, 54369, 58921, 63478, 65478, 67234, 67252, 67254, 69472, 74183, 78123, 78125, 78163, 78165, 78545, 98305, 98325, 105210, 105250, 107632, 107654, 109236, 123072, 123894, 123896, 125010, 125012, 125014, 125016, 125018, 125030, 125032, 125034, 125036, 125038, 125050, 125052, 125054, 125056, 125058, 125070, 125072, 125074, 125076, 125078, 125090, 125092, 125094, 125096, 125098, 125496, 127450, 127454, 127634, 129276, 129438, 129654, 141072, 143658, 145232, 145652, 145656, 145832, 147452, 147456, 147492, 147496, 161050, 161052, 161054, 161056, 161058, 163216, 163258, 163276, 163296, 163832, 163854, 165834, 165874, 165894, 165896, 167274, 167452, 169216, 181476, 183652, 183674, 185652, 187230, 187272, 187278, 187652, 210125, 210147, 210743, 214365, 214369, 218743, 218765, 234123, 234327, 234383, 234385, 234387, 234567, 236145, 236149, 236183, 236189, 236925, 238125, 238143, 238145, 238147, 238325, 238363, 238365, 238563, 250105, 252147, 254369, 256107, 256749, 258363, 274527, 274583, 276561, 276725, 278565, 278943, 278945, 290145, 290521, 290523, 294921, 294965, 294985, 296525, 296585, 296761, 321498, 323676, 325632, 327658, 327674, 329476, 329656, 343072, 343476, 343656, 345472, 345476, 347096, 349492, 349494, 363610, 365472, 365490, 365638, 367236, 367272, 367416, 367472, 367696, 383292, 383856, 389016, 389018, 389278, 389294, 389656, 389672, 389896, 412163, 412923, 432929, 432961, 432967, 436905, 436921, 436927, 438927, 438965, 438967, 438985, 454345, 456921, 458329, 458347, 458381, 458701, 458721, 458723, 458725, 458727, 458729, 458741, 458745, 458747, 458749, 458761, 458763, 458765, 458767, 458769, 458785, 458787, 458789, 470561, 470567, 472349, 472365, 472385, 472389, 472545, 472783, 474543, 474547.

Some larger examples :