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Friedman n.  and  untouchable n.

First numbers that belong to both sets:

216, 1296, 2048, 2500, 2502, 3378, 3972, 4088, 5832, 6880, 9216, 10192, 10368, 10824, 11664, 12060, 12288, 12600, 12762, 12850, 13496, 13824, 13950, 15612, 15618, 15624, 16348, 16374, 16798, 17346, 17482, 17856, 17920, 18432, 19642, 19732, 19734, 21952, 22264, 23392, 23490, 24336, 24390, 24546, 24566, 24576, 24584, 24964, 25872, 28224, 28226, 28476, 28554, 28564, 28674, 28764, 29160, 29768, 31346, 31590, 32744, 33696, 34968, 34986, 34992, 34996, 35726, 35782, 36864, 37668, 38416, 38424, 38856, 39288, 39328, 39366, 39382, 42336, 42898, 43268, 43688, 44676, 45360, 45368, 46368, 46556, 46626, 46632, 46658, 46662, 46672, 47538, 48672, 48756, 49152, 49584, 49896, 49968, 54476, 54642, 54726, 55296, 56628, 58978, 59038, 59048, 59058, 59128, 61440, 62968, 64512, 64522, 64868, 65488, 65536, 67252, 67254, 67950, 69984, 72576, 78126, 78132, 78152, 82372, 82952, 85264, 85358, 86142, 86724, 87382, 92160, 93294, 93312, 93642, 98256, 102528, 102942, 103764, 104260, 104448, 104452, 104816, 104916, 104966, 104968, 104984, 105210, 107632, 109352, 110758, 111538, 112536, 112896, 114242, 114246, 114390, 114684, 115248, 115672, 115968, 116496, 117184, 117396, 117568, 117586, 117624, 117632, 117636, 117642, 117656, 117668, 117686, 117696, 117698, 117786, 117936, 118386, 118440, 118480, 119848, 120006, 120060, 121086, 121576, 123354, 123840, 124362, 124386, 124412, 124416, 124468, 124488, 124792, 124936, 124956, 124968, 125002, 125008, 125010, 125016, 125026, 125034, 125036, 125040, 125046, 125048, 125058, 125060, 125062, 125070, 125072, 125076, 125078, 125080, 125082, 125084, 125092, 125168, 125248, 125316, 125364, 125460, 125496, 125648, 125952, 126000, 126594, 127620, 127744, 128430, 129276, 129438, 129602, 129604, 129648, 129654, 130248, 130942, 131082, 132498, 132596, 133956, 134144, 134462, 134784, 134832, 134960, 135234, 135366, 135828, 136590, 137216, 137632, 137712, 137718, 137728, 137784, 137796, 137798, 137802, 137816, 137826, 137842, 137892, 137924, 139392, 139500, 139676, 139824, 139932, 139962, 139986, 142296, 142376, 143500, 143648, 143748, 143752, 145232, 145314.

Some larger examples :