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balanced prime  and  odious n.

First numbers that belong to both sets:

157, 173, 211, 563, 607, 653, 733, 947, 1123, 1187, 1367, 1747, 1753, 2677, 2963, 3313, 3637, 3733, 4013, 4657, 4993, 5107, 5113, 5393, 5563, 5807, 6073, 6317, 6323, 6367, 6373, 6863, 7823, 7841, 9397, 10253, 10607, 10853, 11411, 11731, 11807, 11903, 11933, 12547, 12583, 13043, 13177, 14543, 14747, 14753, 15161, 15193, 15313, 15467, 15803, 16097, 16427, 16567, 16619, 16937, 16987, 17047, 17327, 17477, 17851, 18217, 18223, 18433, 18731, 19483, 19603, 20123, 20161, 20201, 20347, 20521, 20731, 21911, 22051, 22447, 23893, 24097, 24413, 25391, 26041, 26723, 26987, 28069, 28597, 28927, 29599, 30313, 30637, 30643, 31051, 31327, 32479, 32497, 32993, 34171, 34267, 34877, 35129, 35317, 35911, 36229, 36313, 39113, 39863, 40063, 40093, 40471, 40493, 40933, 40961, 42209, 42397, 42533, 44519, 44543, 44741, 45433, 45497, 46147, 46889, 47303, 47507, 48023, 48661, 49633, 50153, 52957, 53939, 54151, 54443, 54623, 54773, 55207, 56003, 57853, 58067, 58129, 58237, 58573, 58979, 59393, 60617, 61723, 62207, 62633, 63073, 63493, 63527, 63559, 63697, 63703, 64013, 64627, 65413, 67619, 67783, 68099, 68483, 68743, 69067, 71347, 71353, 71843, 72481, 74317, 74471, 74489, 75181, 75577, 76487, 76561, 76673, 77023, 77557, 78173, 78259, 78577, 79187, 79411, 79867, 80251, 80329, 80923, 81001, 81439, 81749, 84053, 84137, 84437, 86981, 87517, 88867, 89003, 89393, 90511, 90703, 91243, 91297, 91303, 91387, 92227, 92363, 92581, 93083, 95917, 95923, 96431, 96763, 97441, 97967, 99823, 100937, 101273, 101483, 103177, 103567, 104021, 104113, 104161, 104953, 105367, 105373, 105607, 105613, 106297, 107581, 107693, 108439, 108637, 108923, 109897, 110459, 110603, 110933, 111521, 111611, 113963, 114083, 114473, 115337, 116107, 116719, 116797, 117437, 117757, 118831, 119563, 121007, 121327, 121447, 123887, 125441, 126211, 127643, 127843, 129049, 129209, 130483, 130687, 130693, 131041, 132511, 132911, 133103, 133571, 133717, 134263, 134333, 134639, 135089, 135301, 135607, 137321, 137393, 137849.

Some larger examples :