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equidigital n.  and  powerful n.

First numbers that belong to both sets:

1, 16, 25, 27, 32, 49, 64, 81, 121, 169, 289, 361, 529, 841, 961, 1000, 1125, 1152, 1225, 1296, 1323, 1372, 1568, 1600, 1728, 1944, 2000, 2025, 2304, 2500, 2592, 2744, 2916, 3087, 3136, 3200, 3375, 3456, 3888, 3969, 4000, 4608, 5000, 5184, 5488, 5625, 5832, 6075, 6125, 6272, 6400, 6912, 7776, 8000, 8575, 8748, 9261, 9604, 10648, 10816, 10952, 11236, 11552, 11979, 12321, 13225, 13448, 13456, 13689, 13924, 14161, 14283, 14792, 14884, 15125, 15129, 15376, 15488, 16641, 16928, 17576, 17672, 17689, 17956, 18432, 18496, 19652, 19773, 19881, 20164, 21025, 21125, 21296, 21316, 21632, 21904, 22472, 22707, 23104, 23409, 24025, 24964, 25281, 25600, 25921, 25947, 26896, 26912, 27436, 27556, 27648, 27848, 29241, 29403, 29584, 29768, 30752, 30976, 31329, 31684, 33275, 33489, 33856, 34225, 35152, 35344, 35912, 35937, 36125, 36864, 36963, 36992, 37636, 39304, 40328, 40401, 41067, 41209, 41503, 42025, 42592, 42632, 42849, 43264, 43808, 44217, 44944, 45125, 45369, 45387, 46208, 46225, 47089, 47961, 48668, 49923, 49928, 50176, 51200, 53792, 53824, 54872, 54925, 55112, 55225, 55296, 55696, 56169, 57967, 58564, 59168, 59319, 59536, 59643, 61504, 61731, 61952, 62001, 63368, 65219, 66125, 67081, 67712, 68121, 70225, 70227, 70304, 70688, 71289, 71824, 73728, 73984, 75272, 75625, 75843, 77841, 78608, 80656, 82369, 82944, 84681, 85184, 85264, 86528, 87025, 87616, 87723, 88209, 89888, 90601, 92416, 93025, 93987, 97336, 97556, 99127, 99856, 101761, 102152, 103041, 104329, 106276, 106929, 108000, 111132, 111556, 112500, 112896, 114921, 115200, 116281, 119716, 121500, 122500, 123904, 127008, 128164, 129032, 129600, 131044, 135000, 137200, 137288, 142129, 142884, 144000, 145161, 145800, 145924, 148176, 148996, 150152, 152881, 154449, 154568, 155236, 156800, 158404, 162000, 162409, 163216, 165375, 165649, 168921, 169344, 169744, 171500, 172800, 172872, 173056, 173889, 177608, 178084, 180000, 182408, 183184, 190096, 190512, 190969, 194400.

Some larger examples :