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arithmetic n.  and  Ruth-Aaron pair

First numbers that belong to both sets:

5, 6, 15, 49, 77, 78, 105, 125, 126, 153, 154, 369, 492, 493, 714, 715, 949, 1330, 1331, 1520, 1683, 1862, 2107, 2108, 2491, 2492, 2783, 2784, 3248, 4185, 4186, 4191, 4192, 5405, 5406, 5561, 5959, 6556, 6557, 6811, 6812, 6868, 8280, 8281, 8463, 8855, 12351, 12726, 12727, 13775, 14587, 14588, 16932, 16933, 17080, 17081, 18491, 18655, 18656, 20451, 21183, 24024, 24432, 24433, 24881, 24895, 25839, 25840, 26642, 26643, 28448, 28449, 28809, 28810, 33019, 33020, 35457, 37829, 37881, 37882, 40081, 41261, 41262, 42625, 43215, 43680, 44831, 44832, 44891, 47544, 47545, 48203, 49240, 49241, 52554, 52555, 53193, 57075, 61761, 63345, 63426, 63427, 63665, 63666, 64232, 68264, 68265, 68949, 68950, 70356, 72500, 72501, 75141, 79118, 79119, 81175, 89979, 89980, 95709, 95710, 98119, 98120, 98645, 99163, 99164, 106799, 106800, 106893, 106894, 109939, 109940, 110607, 112079, 113283, 118657, 128774, 128775, 132209, 141340, 141341, 141488, 143622, 143623, 148215, 148216, 154842, 154843, 155111, 155112, 155708, 155709, 172773, 172774, 173162, 173163, 174096, 174097, 180359, 180360, 183446, 183447, 189081, 197657, 197658, 204258, 204259, 204610, 204611, 208581, 208582, 215391, 215392, 216293, 216294, 224105, 224257, 228599, 230125, 231284, 231285, 235587, 244035, 244595, 244596, 244853, 244854, 247456, 248109, 248110, 250995, 250996, 261019, 261020, 261569, 263327, 263328, 263628, 263629, 265112, 265113, 268656, 268657, 268715, 278277, 278278, 289316, 289317, 292454, 292455, 294335, 296111, 296112, 296751, 304325, 304326, 319428, 319429, 329988, 329989, 332994, 332995, 333802, 346692, 346693, 350599, 351076, 361063, 370348, 370349, 373703, 373704, 380438, 380439, 380963, 380964, 381728, 381729, 384312, 384313, 388451, 394855, 394856, 404973, 404974, 407835, 407836, 416120, 417162, 417163, 417164, 421820, 430604, 430605, 436932, 436933, 437715, 437716, 441211, 441212, 445305, 453474, 453475, 457899, 461825, 466773, 466774, 467302, 467303, 470120, 477092, 477093, 493923, 501830, 501831, 508563, 508564, 509851, 509852, 525224, 525225.

Some larger examples :