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Friedman n.  and  happy n.

First numbers that belong to both sets:

736, 1285, 4096, 4536, 10251, 10521, 10575, 11025, 12101, 12102, 12105, 12108, 12167, 12321, 12384, 12546, 12762, 12843, 12850, 12996, 13822, 13824, 13825, 13842, 14129, 14256, 14641, 14645, 15246, 15562, 15622, 15624, 15625, 15628, 15641, 15642, 15645, 15679, 15688, 15697, 16225, 16245, 16272, 16448, 16759, 17384, 18265, 18432, 19392, 19592, 19629, 19692, 19693, 19736, 20485, 21586, 21843, 24375, 24546, 24564, 24576, 26348, 26754, 27654, 27889, 28561, 29519, 29524, 29525, 29584, 32744, 32782, 32815, 32851, 34425, 34991, 35933, 36549, 37814, 38424, 38637, 39283, 39313, 39328, 39358, 39362, 39382, 39456, 39784, 41468, 41472, 42025, 43268, 45360, 45369, 46593, 46630, 46645, 46663, 46668, 46875, 47652, 48756, 48757, 52168, 52429, 52498, 53245, 54369, 54432, 54642, 54726, 54768, 59049, 59056, 59094, 59129, 59409, 63945, 64036, 64512, 65478, 65487, 67254, 70225, 73984, 74183, 82372, 82755, 83521, 87382, 102051, 102382, 102384, 102510, 102912, 103824, 104328, 104375, 105021, 105075, 105210, 105264, 105624, 105625, 105750, 106929, 108135, 108241, 109962, 110250, 111537, 112895, 112923, 114395, 114687, 115934, 115968, 116317, 116478, 116625, 117316, 117466, 117568, 117572, 117586, 117596, 117613, 117623, 117632, 117640, 117646, 117648, 117658, 117659, 117664, 117684, 117685, 117695, 117739, 118259, 118567, 120384, 120762, 120843, 121203, 123084, 123265, 123648, 123840, 123897, 123935, 124362, 124368, 124386, 124479, 124632, 124993, 125001, 125007, 125008, 125010, 125011, 125026, 125038, 125046, 125056, 125062, 125064, 125065, 125068, 125070, 125080, 125083, 125086, 125099, 125364, 125460, 126025, 126027, 127449, 127587, 127620, 128403, 128430, 128500, 129609, 130248, 130679, 131082, 131850, 132499, 132565, 132568, 132625, 132645, 132652, 132654, 132655, 132658, 132685, 132797, 133125, 133815, 134375, 134456, 134462, 134465, 134832, 135625, 135828, 136079, 136525, 137216, 137344, 137635, 137719, 137728, 137729, 137768, 137782, 137784, 137788, 137791, 137792, 137892, 137928, 137955, 137964, 138240, 138624, 138784, 140625, 141345, 142795, 142975, 143357, 143465, 145314, 145656.

Some larger examples :