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balanced n.  and  Duffinian n.

First numbers that belong to both sets:

9, 21, 35, 49, 50, 75, 169, 201, 209, 225, 527, 535, 539, 551, 565, 578, 583, 589, 595, 597, 611, 625, 651, 676, 689, 707, 713, 721, 722, 737, 779, 781, 793, 803, 805, 817, 835, 837, 841, 849, 865, 899, 901, 905, 913, 961, 968, 1922, 2107, 2127, 2135, 2155, 2163, 2165, 2169, 2191, 2199, 2215, 2219, 2227, 2233, 2253, 2261, 2289, 2327, 2331, 2361, 2379, 2387, 2405, 2439, 2451, 2453, 2469, 2501, 2513, 2575, 2587, 2599, 2603, 2605, 2611, 2613, 2631, 2635, 2645, 2649, 2665, 2695, 2709, 2723, 2737, 2738, 2757, 2761, 2785, 2827, 2885, 2916, 2947, 2977, 3087, 3099, 3101, 3117, 3125, 3143, 3149, 3155, 3161, 3171, 3173, 3207, 3211, 3225, 3235, 3241, 3267, 3269, 3281, 3297, 3341, 3349, 3353, 3365, 3369, 3397, 3401, 3409, 3425, 3459, 3473, 3591, 3603, 3605, 3619, 3653, 3665, 3681, 3715, 3721, 3745, 3843, 3857, 3873, 3905, 6713, 6721, 6767, 6773, 6775, 6799, 6881, 6989, 7025, 7031, 7041, 7091, 7097, 7099, 7115, 7117, 7123, 7153, 7163, 7203, 7223, 7225, 7231, 7241, 7255, 7267, 7273, 7275, 7361, 7367, 7397, 7475, 7511, 7513, 7525, 7527, 7595, 7715, 7751, 7799, 7801, 7811, 7835, 7837, 7849, 7851, 7865, 7913, 7943, 7945, 7957, 7959, 7967, 8001, 8063, 8071, 8125, 8135, 8159, 8165, 8173, 8197, 8203, 8213, 8215, 8227, 8229, 8239, 8245, 8247, 8255, 8279, 8285, 8303, 8309, 8315, 8321, 8327, 8335, 8337, 8351, 8375, 8381, 8399, 8407, 8411, 8413, 8427, 8435, 8437, 8453, 8479, 8503, 8507, 8509, 8551, 8553, 8557, 8569, 8571, 8591, 8593, 8603, 8605, 8615, 8621, 8633, 8643, 8651, 8659, 8661, 8665, 8671, 8675, 8695, 8697, 8703, 8735, 8751, 8759, 8765, 8791, 8797, 8801, 8825, 8855, 8859, 8875, 8885, 8891, 8903, 8909, 8915, 8917.

Some larger examples :