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balanced n.  and  Ruth-Aaron pair

First numbers that belong to both sets:

9, 15, 49, 50, 78, 153, 154, 714, 2107, 3249, 8855, 13776, 16933, 28449, 33019, 37829, 43680, 48204, 52554, 172773, 172774, 173163, 183446, 189080, 189081, 204610, 228600, 230124, 244036, 244854, 250996, 332994, 381728, 388450, 404974, 416120, 417164, 437716, 441212, 457900, 467302, 537917, 540280, 557535, 563390, 565823, 576984, 583462, 643467, 665322, 671020, 685228, 693610, 694430, 694431, 732199, 742508, 773313, 788671, 801295, 820601, 827896, 839376, 856607, 878898, 890126, 901748, 906020, 909624, 927775, 932035, 961573, 977281, 1019425, 1019426, 1040403, 2179519, 2305396, 2314663, 2314664, 2339046, 2373453, 2390482, 2407320, 2493799, 2527020, 2538533, 2538534, 2559672, 2639703, 2649041, 2649042, 2701657, 2701669, 2776184, 2802297, 2802298, 2834767, 2849042, 2923921, 2992220, 3003468, 3026019, 3032760, 3056977, 3164524, 3165639, 3205916, 3206342, 3223789, 3230073, 3230074, 3265805, 3265806, 3369764, 3394016, 3402312, 3414889, 3423436, 3429081, 3457497, 3465088, 3476964, 3524496, 3570732, 3584053, 3584195, 3702112, 3749428, 3816036, 3835132, 3912336, 3950119, 3978261, 4010181, 4010182, 4010552, 4055618, 4057122, 4076328, 4334144, 4481982, 4854877, 4958989, 5096736, 5103875, 5152756, 5213349, 5483189, 5544715, 5800392, 5800393, 5974320, 6029172, 6066072, 6126439, 6152293, 6157664, 6157665, 6262319, 6440240, 6445655, 6473349, 6523100, 6529003, 6529004, 6531669, 6540897, 6599760, 6668948, 7094932, 7304433, 7387680, 7689303, 8062119, 8062815, 8074785, 8108057, 8251407, 8407619, 8470480, 8470481, 8471039, 8533851, 8559419, 8608668, 8746897, 8800733, 8818147, 8864700, 8976852, 9018427, 9171650, 9200861, 9203335, 9249750, 9251927, 9263525, 9263526, 9329191, 9366030, 9532061, 9532062, 9820609, 9940610, 9981936, 10050851, 10064295, 10079796, 10108808, 10122763, 10133251, 10201268, 10229101, 10229102, 10321503, 10389686, 10876547, 10876548, 10902188, 10906005, 11054808, 11162348, 11219180, 11240761, 11261040, 11305431, 11322923, 11341213, 11371367, 11390267, 11391139, 11436487, 11480288, 11587931, 11597909, 11599955, 11661672, 11690730, 11793552, 11795373, 11802482, 11819804, 11885057, 11930049, 11956549, 11956550, 12117170, 12120201, 12130805, 12146204, 12202078, 12282920, 12288569, 12288570, 12369045, 12473847, 12735885, 12757492, 12812765, 12817637, 13087239, 13092907, 13160028, 13227410, 13237024, 13247551, 13346465, 13470965, 13536098, 13552450, 13790985, 13847706, 13867665, 13902834, 13902835, 13994694, 13999456, 14030341, 14185161.

Some larger examples :