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balanced prime  and  happy n.

First numbers that belong to both sets:

263, 563, 653, 1511, 2417, 2903, 2963, 3313, 3637, 6073, 6367, 6373, 7841, 10607, 12583, 12841, 13177, 13457, 14753, 16097, 16937, 17483, 18223, 18341, 23893, 25621, 26177, 26393, 30097, 30103, 30313, 30637, 30643, 30977, 32083, 36067, 40111, 40471, 42683, 44041, 44519, 44543, 47507, 48259, 50417, 51461, 52529, 54151, 54443, 55207, 55843, 56003, 58543, 60083, 60337, 62633, 63073, 63559, 63703, 67169, 67619, 68099, 71843, 74489, 75787, 76913, 78577, 80251, 82463, 82787, 83003, 84137, 87517, 87547, 90703, 91303, 92003, 92363, 98473, 99823, 101483, 103177, 104113, 104207, 104723, 107693, 110681, 111509, 113921, 114001, 114083, 114901, 115337, 120823, 121327, 122833, 126317, 130343, 131041, 132911, 133571, 137131, 141067, 142019, 143159, 144439, 145037, 145879, 146701, 148867, 150407, 151201, 151687, 152623, 152947, 154073, 162853, 165443, 168737, 171091, 173183, 176207, 176401, 177301, 179969, 182279, 184837, 186253, 186877, 187367, 188159, 188317, 188437, 200177, 201151, 202387, 203227, 206221, 206483, 211193, 213223, 214463, 216731, 217163, 221233, 230227, 235723, 238207, 243857, 250919, 252163, 253969, 254179, 255517, 258539, 260363, 265123, 265813, 267907, 270223, 270967, 273187, 278617, 279137, 280933, 287173, 289999, 290033, 292477, 294787, 297607, 298093, 300667, 301039, 302779, 302977, 302983, 304217, 304517, 305821, 307079, 312407, 316891, 318431, 318817, 320693, 321721, 327337, 329717, 331171, 334177, 336157, 336983, 341191, 344237, 344243, 344423, 346349, 351047, 355507, 355847, 356299, 357683, 361357, 363767, 366133, 370603, 373693, 375163, 376477, 376583, 381401, 381817, 382807, 389369, 390353, 392531, 393103, 396091, 396443, 397217, 399899, 401237, 402071, 408223, 412019, 415409, 418459, 424163, 428759, 430499, 432743, 434243, 437401, 440543, 441563, 441613, 443237, 443243, 445877, 450077, 451337, 458651, 467101, 470501, 473513, 476243, 477329, 478099, 479581, 479777, 480107, 480749, 483557, 484079, 485941, 487463, 494693, 495401, 496211, 496487, 496493, 497297, 508097, 509563, 509653, 510299, 511001, 511261, 513047, 513347, 517073, 521497, 530983, 532193, 532391, 535511, 539141, 540173, 551569.

Some larger examples :