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arithmetic n.  and  Ulam n.

First numbers that belong to both sets:

1, 3, 6, 11, 13, 38, 47, 53, 57, 62, 69, 77, 87, 97, 99, 102, 114, 126, 131, 138, 145, 155, 177, 182, 189, 197, 206, 209, 219, 221, 236, 238, 241, 253, 258, 260, 273, 282, 309, 319, 339, 341, 356, 358, 382, 390, 402, 409, 414, 429, 431, 434, 451, 456, 483, 485, 497, 502, 524, 546, 566, 568, 585, 602, 605, 607, 612, 627, 646, 668, 673, 685, 690, 695, 734, 739, 751, 781, 783, 798, 820, 849, 861, 864, 893, 905, 949, 983, 991, 1020, 1023, 1025, 1030, 1032, 1035, 1037, 1052, 1079, 1081, 1101, 1103, 1125, 1155, 1157, 1167, 1169, 1191, 1208, 1230, 1257, 1311, 1313, 1335, 1338, 1340, 1355, 1387, 1389, 1406, 1428, 1431, 1433, 1462, 1465, 1470, 1489, 1509, 1531, 1550, 1553, 1604, 1641, 1643, 1646, 1660, 1707, 1709, 1721, 1724, 1748, 1765, 1770, 1790, 1814, 1834, 1836, 1853, 1902, 1919, 1941, 1946, 1966, 1985, 2010, 2012, 2054, 2090, 2093, 2095, 2115, 2117, 2134, 2156, 2247, 2249, 2252, 2254, 2327, 2332, 2354, 2371, 2393, 2418, 2445, 2447, 2462, 2464, 2481, 2484, 2486, 2513, 2525, 2550, 2574, 2581, 2589, 2613, 2618, 2630, 2633, 2635, 2660, 2662, 2674, 2721, 2723, 2750, 2787, 2789, 2811, 2814, 2831, 2833, 2897, 2899, 2919, 2921, 2985, 2987, 3029, 3031, 3038, 3041, 3043, 3065, 3068, 3070, 3085, 3090, 3092, 3107, 3109, 3131, 3153, 3205, 3207, 3214, 3217, 3219, 3236, 3239, 3261, 3263, 3290, 3305, 3371, 3373, 3390, 3393, 3395, 3415, 3417, 3451, 3454, 3473, 3476, 3495, 3525, 3527, 3547, 3549, 3591, 3593, 3605, 3622, 3625, 3630, 3649, 3669, 3671, 3691, 3723, 3725, 3740, 3742, 3759, 3762, 3764, 3806, 3808, 3825, 3886, 3916, 3918, 3930, 3952, 3962, 3974, 3991, 4038.

Some larger examples :