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interprime n.  and  untouchable n.

First numbers that belong to both sets:

120, 246, 288, 324, 342, 426, 552, 714, 792, 1044, 1078, 1254, 1404, 1476, 1632, 1728, 1806, 1884, 1922, 1962, 2022, 2058, 2148, 2322, 2364, 2430, 2490, 2568, 2640, 2724, 2736, 2760, 2922, 2946, 3114, 3366, 3576, 3588, 3600, 3684, 3774, 3786, 3812, 3828, 4168, 4206, 4224, 4248, 4312, 4344, 4368, 4382, 4472, 4488, 4500, 4612, 4668, 4712, 4742, 4896, 4914, 4962, 4980, 5016, 5160, 5184, 5218, 5316, 5328, 5366, 5492, 5544, 5664, 5796, 6060, 6084, 6168, 6282, 6384, 6460, 6538, 6684, 6726, 7050, 7074, 7140, 7168, 7200, 7302, 7360, 7402, 7468, 7512, 7554, 7614, 7734, 7774, 7860, 7892, 8028, 8046, 8064, 8154, 8280, 8382, 8404, 8532, 8568, 8616, 8770, 8826, 8844, 9144, 9302, 9408, 9450, 9504, 9684, 9708, 9758, 9894, 9958, 10074, 10146, 10308, 10326, 10578, 10620, 10700, 10762, 10776, 10794, 10944, 11064, 11154, 11166, 11226, 11340, 11376, 11430, 11694, 11760, 11964, 12060, 12258, 12360, 12444, 12522, 12680, 12708, 12772, 12932, 12992, 13056, 13202, 13224, 13320, 13374, 13506, 13530, 13560, 13572, 13584, 13674, 13716, 13818, 13926, 13948, 14020, 14076, 14166, 14232, 14312, 14334, 14484, 14526, 14676, 14706, 14790, 14934, 15024, 15126, 15282, 15294, 15340, 15396, 15420, 15456, 15504, 15592, 15624, 15744, 15780, 15868, 15948, 15996, 16162, 16240, 16310, 16716, 17002, 17130, 17274, 17308, 17346, 17604, 17616, 17742, 17886, 17916, 17934, 18054, 18108, 18138, 18186, 18388, 18420, 18498, 18568, 18696, 18780, 18966, 19060, 19104, 19172, 19326, 19410, 19452, 19704, 19770, 19830, 19848, 19902, 19932, 19968, 20038, 20136, 20192, 20364, 20436, 20538, 20700, 20712, 20892, 20912, 20934, 20972, 20992, 21130, 21216, 21298, 21332, 21426, 21582, 21594, 21692, 21762, 21780, 21810, 21830, 21846, 21876, 21902, 21920, 21952, 22008, 22180, 22358, 22386, 22492, 22506, 22536, 22660, 22704, 22760, 22914, 22952, 23034, 23218, 23304.

Some larger examples :