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balanced n.  and  gapful n.

First numbers that belong to both sets:

108, 120, 135, 150, 154, 165, 170, 180, 195, 198, 225, 240, 260, 396, 462, 500, 550, 572, 583, 600, 660, 682, 781, 880, 990, 1070, 1110, 1190, 1210, 1470, 1490, 1670, 1730, 1790, 1830, 2222, 2236, 2268, 2275, 2331, 2349, 2375, 2420, 2424, 2457, 2460, 2520, 2544, 2575, 2620, 2662, 2680, 2725, 2740, 2772, 2800, 2860, 2882, 2886, 2900, 2904, 2920, 2961, 2968, 2980, 2992, 3115, 3150, 3180, 3185, 3270, 3276, 3300, 3395, 3465, 3480, 3552, 3605, 3636, 3660, 3745, 3780, 4000, 4032, 6695, 6767, 7189, 7275, 7881, 8245, 8415, 8585, 8692, 8989, 9450, 9785, 9898, 9900, 10005, 10010, 10021, 10023, 10035, 10040, 10060, 10065, 10096, 10130, 10131, 10152, 10160, 10180, 10241, 10275, 10279, 10350, 10387, 10395, 10413, 10425, 10458, 10460, 10469, 10470, 10510, 10545, 10550, 10570, 10571, 10575, 10610, 10635, 10638, 10650, 10659, 10665, 10673, 10680, 10700, 10724, 10725, 10728, 10736, 10755, 10791, 10803, 10810, 10812, 10830, 10860, 10872, 10901, 10905, 10908, 10930, 10932, 10950, 10980, 10992, 11011, 11030, 11039, 11050, 11052, 11060, 11063, 11067, 11085, 11090, 11112, 11120, 11145, 11160, 11170, 11172, 11190, 11200, 11216, 11220, 11229, 11232, 11237, 11265, 11310, 11341, 11350, 11354, 11355, 11370, 11380, 11385, 11415, 11420, 11430, 11448, 11453, 11470, 11475, 11480, 11490, 11535, 11560, 11561, 11564, 11565, 11570, 11580, 11590, 11595, 11610, 11620, 11660, 11696, 11713, 11715, 11720, 11770, 11790, 11805, 11820, 11832, 11840, 11843, 11850, 11860, 11870, 11880, 11910, 11914, 11940, 11952, 11970, 11984, 12000, 12050, 12072, 12075, 12080, 12087, 12100, 12130, 12165, 12168, 12176, 12179, 12192, 12195, 12233, 12270, 12280, 12285, 12300, 12315, 12350, 12375, 12405, 12410, 12427, 12435, 12441, 12470, 12474, 12493, 12495, 12530, 12559, 12560, 12585, 12590, 12597, 12615, 12630, 12645, 12650, 12660, 12700, 12710, 12720, 12735, 12753.

Some larger examples :