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amenable n.  and  balanced prime

First numbers that belong to both sets:

5, 53, 157, 173, 257, 373, 593, 653, 733, 977, 1753, 2417, 2677, 3313, 3637, 3733, 4013, 4409, 4457, 4597, 4657, 4993, 5113, 5393, 6073, 6317, 6373, 6977, 7841, 8117, 8713, 9397, 9473, 10253, 10657, 10853, 11497, 11933, 12497, 12653, 12841, 12973, 13177, 13457, 14753, 15161, 15193, 15313, 15797, 15913, 16097, 16937, 17477, 18217, 18341, 18433, 19477, 19577, 19949, 20161, 20201, 20521, 21661, 21997, 22073, 23333, 23801, 23893, 24077, 24097, 24413, 24677, 25621, 25673, 26041, 26177, 26393, 26693, 27773, 28069, 28597, 29173, 29269, 29333, 29873, 30097, 30313, 30449, 30637, 30977, 32497, 32993, 33113, 33857, 34673, 34877, 35129, 35317, 35753, 36229, 36313, 36677, 36821, 37957, 39113, 40093, 40493, 40933, 40961, 41017, 41513, 42209, 42397, 42533, 43189, 44041, 44741, 45433, 45497, 45757, 46889, 47869, 48533, 48661, 49069, 49633, 50153, 50417, 50441, 51413, 51461, 52529, 52957, 53617, 54673, 54773, 55793, 56093, 56149, 56437, 56773, 56957, 57853, 58129, 58237, 58573, 58693, 59113, 59393, 60133, 60337, 60509, 60617, 61553, 61637, 62213, 62597, 62633, 63073, 63493, 63697, 64013, 65413, 65557, 65957, 66809, 67169, 71353, 71993, 72481, 72733, 73477, 74317, 74489, 74897, 75181, 75217, 75533, 75577, 75641, 76561, 76673, 76913, 77081, 77557, 78173, 78577, 78797, 79973, 80329, 80917, 81001, 81077, 81689, 81749, 81937, 82493, 82793, 83437, 84053, 84137, 84437, 85049, 85121, 85837, 86137, 86981, 87517, 89113, 89393, 91297, 92581, 93053, 93377, 94433, 95273, 95917, 96001, 96377, 97441, 98473, 98597, 100937, 101273, 103177, 103669, 104021, 104053, 104113, 104161, 104717, 104953, 105373, 105613, 105977, 106297, 106681, 107581, 107693, 108217, 108637, 109897, 110681, 110813, 110933, 111509, 111521, 111773, 112181, 112297, 113117, 113921, 114001, 114473, 114901, 115337, 115757, 116797, 117437, 117757, 119033, 119557, 120097, 120557, 120817, 121013, 121421, 121553, 122273, 122833, 123433, 125441, 125737, 126317, 127733, 128153, 128449, 129049, 129209, 130693.

Some larger examples :