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de Polignac n.  and  emirpimes

First numbers that belong to both sets:

1211, 1243, 1807, 1927, 1969, 1985, 2231, 2263, 2279, 2429, 2669, 3215, 3431, 3505, 3665, 3817, 3985, 4151, 4195, 4573, 4781, 5143, 5617, 5917, 6001, 6065, 6283, 6403, 6463, 6821, 7169, 7199, 7289, 7343, 7387, 7431, 7799, 7807, 7913, 8141, 8159, 8399, 8587, 9569, 9641, 9809, 9959, 10021, 10027, 10237, 10327, 10447, 10483, 10553, 10873, 11015, 11141, 11207, 11219, 11231, 11335, 11729, 11771, 11857, 11993, 12223, 12233, 12877, 13393, 13589, 13897, 13973, 14023, 14227, 14383, 14531, 14585, 14611, 14681, 14765, 15127, 15247, 15419, 15701, 15779, 15943, 15997, 16013, 16109, 16213, 16409, 16499, 16507, 16601, 16645, 16753, 16783, 16865, 16867, 17039, 17083, 17089, 17147, 17229, 17305, 17369, 17429, 17593, 17735, 17773, 17849, 17887, 18033, 18155, 18607, 18613, 18889, 18895, 18897, 19361, 20321, 20381, 20467, 20617, 20623, 20957, 22327, 22331, 22361, 22633, 22657, 22663, 24131, 24461, 24491, 24569, 24641, 24731, 24833, 24899, 24913, 24929, 26413, 26473, 26531, 26567, 26933, 28169, 28207, 28331, 28333, 28781, 28799, 30281, 30587, 30599, 30629, 30755, 30787, 31061, 31499, 31709, 31843, 32047, 32231, 32639, 33221, 33253, 33373, 33389, 33667, 33685, 33689, 33763, 34003, 34009, 34169, 34829, 34889, 34931, 34973, 35263, 35459, 35699, 35909, 35947, 35957, 36035, 36503, 36581, 36667, 36719, 37237, 37477, 37669, 37753, 37837, 37841, 37931, 38089, 38137, 38413, 38429, 38497, 38999, 39007, 39131, 39287, 39539, 39817, 41639, 41939, 43145, 43289, 43367, 43619, 43679, 43739, 43859, 43873, 43957, 45487, 45581, 45803, 45851, 47231, 47327, 47429, 47455, 47833, 47923, 49319, 49717, 49721, 49829, 50339, 50429, 50489, 50579, 50737, 50801, 51011, 51041, 51083, 51277, 51697, 51703, 51739, 51761, 51823, 51959, 53195, 53321, 53347, 53351, 53399, 53561, 53711, 53969, 54007, 54031, 54097, 54161, 54491, 54569, 54707, 54809, 55169, 55273, 55423, 55435, 55483, 56191, 56219, 56291, 56293, 56411, 56429, 56609, 56669.

Some larger examples :