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happy n.  and  heptagonal n.

First numbers that belong to both sets:

1, 7, 469, 874, 970, 1177, 1288, 3010, 3367, 3553, 4558, 4774, 4995, 5452, 7480, 8323, 20115, 21022, 21483, 22420, 24354, 24850, 25857, 27405, 34633, 37638, 42055, 45360, 46036, 51624, 52345, 54538, 63760, 76300, 78943, 79834, 92833, 93799, 96727, 101707, 117397, 137710, 158382, 166023, 183196, 188650, 205492, 211266, 221563, 226051, 233631, 235162, 236698, 244453, 253924, 265201, 285103, 291897, 295324, 316306, 325261, 345402, 349129, 360430, 362331, 379665, 389470, 417589, 448804, 472410, 489958, 492174, 510082, 516880, 523723, 526014, 528310, 546858, 575280, 584914, 594628, 601966, 619263, 670033, 677821, 698809, 722803, 736308, 777573, 854685, 857611, 887146, 911134, 920212, 932386, 938503, 1029447, 1032658, 1055275, 1088010, 1091311, 1097928, 1114558, 1131313, 1134679, 1138050, 1151584, 1154980, 1175461, 1199583, 1206520, 1213477, 1227451, 1262736, 1302127, 1320232, 1338462, 1342123, 1371591, 1386445, 1408876, 1458094, 1481095, 1484946, 1559065, 1594804, 1610818, 1655269, 1687977, 1804975, 1809226, 1864944, 1873591, 1882258, 1943487, 1947898, 1956735, 1974469, 1978915, 1983366, 2001220, 2091690, 2264332, 2273859, 2283406, 2297764, 2350795, 2424085, 2433942, 2498500, 2508507, 2513518, 2548735, 2619904, 2650705, 2661012, 2738952, 2759926, 2780980, 2908984, 2952292, 2984983, 3012363, 3028851, 3039868, 3045384, 3061962, 3078585, 3145527, 3162375, 3196206, 3207523, 3224536, 3275845, 3287302, 3293038, 3316032, 3508785, 3520642, 3592204, 3622234, 3628255, 3634281, 3670542, 3713074, 3719170, 3805039, 3866974, 3910626, 3923143, 3979717, 4043052, 4049413, 4062150, 4106887, 4113298, 4164766, 4171222, 4236057, 4294836, 4354020, 4367227, 4446889, 4534002, 4567732, 4656015, 4676508, 4717629, 4897900, 4947012, 5017597, 5060188, 5189041, 5246829, 5275843, 5326810, 5400045, 5458993, 5540569, 5555466, 5592796, 5630251, 5652784, 5682898, 5766124, 5819401, 5872923, 5926690, 5949808, 6112894, 6159895, 6199200, 6341733, 6373627, 6550474, 6566671, 6656112, 6729741, 6770821, 6795529, 6820282, 6886510, 6894811, 6911428, 6919744, 6986452, 7205463, 7265005, 7359066, 7566390, 7575091, 7583797, 7618671, 7662376, 7732564, 7820749, 7829595, 7980742, 8097300, 8241916, 8332951, 8396973, 8433667, 8711422, 8776879, 8861398, 8880235, 8908528, 8936866, 8974720, 9117385, 9184347, 9203524, 9213120, 9309355, 9319006, 9357660, 9396394, 9444924, 9464371, 9474102, 9610861, 9758476, 9768357, 9837664, 9877378, 9987004.