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balanced n.  and  equidigital n.

First numbers that belong to both sets:

2, 10, 11, 15, 19, 21, 35, 37, 41, 49, 135, 139, 141, 142, 147, 149, 163, 166, 169, 177, 178, 197, 201, 384, 541, 557, 563, 569, 587, 601, 613, 617, 647, 653, 659, 661, 677, 709, 787, 809, 841, 929, 961, 1022, 1138, 1142, 1202, 1294, 1346, 1358, 1366, 1454, 1478, 1646, 1654, 1726, 1766, 1922, 1934, 1982, 2095, 2103, 2107, 2127, 2135, 2141, 2155, 2157, 2165, 2191, 2199, 2203, 2215, 2219, 2221, 2229, 2251, 2253, 2281, 2319, 2333, 2347, 2349, 2357, 2361, 2375, 2381, 2389, 2393, 2403, 2417, 2430, 2443, 2467, 2469, 2473, 2481, 2501, 2513, 2589, 2605, 2611, 2617, 2631, 2643, 2645, 2649, 2659, 2661, 2662, 2673, 2699, 2701, 2707, 2713, 2723, 2729, 2738, 2744, 2757, 2785, 2823, 2835, 2837, 2841, 2851, 2857, 2883, 2885, 2897, 2913, 2916, 2947, 2949, 2953, 3087, 3095, 3101, 3115, 3143, 3149, 3155, 3221, 3235, 3241, 3269, 3347, 3353, 3365, 3395, 3397, 3402, 3409, 3461, 3521, 3595, 3625, 3665, 3715, 3845, 3888, 3969, 4000, 6719, 6791, 6793, 6857, 6875, 6881, 6883, 6911, 6947, 6959, 6983, 6991, 7001, 7013, 7027, 7031, 7039, 7151, 7187, 7193, 7243, 7247, 7369, 7433, 7477, 7487, 7499, 7517, 7559, 7607, 7703, 7793, 7823, 7841, 7877, 7949, 7993, 8069, 8087, 8089, 8117, 8123, 8125, 8147, 8161, 8219, 8287, 8297, 8311, 8317, 8363, 8375, 8423, 8427, 8429, 8431, 8443, 8521, 8527, 8537, 8539, 8543, 8563, 8597, 8599, 8627, 8629, 8633, 8641, 8647, 8677, 8681, 8689, 8783, 8807, 8819, 8821, 8861, 8875, 8933, 8969, 8971, 8999, 9011, 9013, 9043, 9049, 9059, 9109, 9125, 9137, 9157, 9161, 9173, 9209, 9257, 9277, 9281, 9293, 9319, 9323, 9337, 9371, 9397, 9403, 9419, 9421, 9433, 9511, 9539, 9547, 9551, 9613, 9619, 9649, 9697, 9743.

Some larger examples :