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alternating n.  and  weak prime

First numbers that belong to both sets:

3, 7, 23, 43, 47, 61, 83, 89, 103, 109, 167, 181, 349, 383, 389, 503, 509, 523, 547, 709, 743, 761, 941, 983, 2143, 2161, 2341, 2383, 2543, 2909, 2927, 4129, 4363, 4523, 4549, 4567, 4703, 4723, 4789, 4909, 4943, 4969, 6101, 6121, 6301, 6329, 6343, 6361, 6529, 6581, 6703, 6709, 6763, 6781, 6949, 6961, 6983, 8101, 8123, 8329, 8369, 8389, 8527, 8543, 8581, 8741, 8761, 8783, 8941, 10103, 10141, 10163, 10169, 10181, 10303, 10321, 10343, 10369, 10501, 10567, 10789, 10909, 12101, 12109, 12149, 12163, 12301, 12329, 12347, 12503, 12527, 12541, 12589, 12703, 12721, 12743, 12763, 12923, 12983, 14107, 14303, 14323, 14327, 14347, 14389, 14503, 14563, 14723, 14741, 14783, 14929, 14969, 14983, 16103, 16141, 16189, 16349, 16363, 16369, 16381, 16529, 16703, 16747, 16763, 16903, 16943, 16981, 18121, 18149, 18503, 18523, 18541, 18587, 18701, 18749, 30169, 30187, 30323, 30347, 30367, 30707, 30727, 30763, 30781, 30941, 30949, 30983, 32143, 32303, 32309, 32323, 32327, 32363, 32381, 32507, 32563, 32587, 32783, 32789, 32941, 32987, 34129, 34147, 34183, 34303, 34327, 34369, 34381, 34501, 34549, 34703, 34729, 34747, 34763, 34781, 34949, 34963, 34981, 36109, 36161, 36343, 36389, 36529, 36563, 36587, 36721, 36749, 36767, 36781, 36901, 36923, 36947, 38189, 38303, 38321, 38329, 38501, 38561, 38569, 38729, 38749, 38903, 38923, 50101, 50123, 50341, 50363, 50387, 50503, 50543, 50723, 50789, 50929, 52127, 52163, 52183, 52189, 52301, 52321, 52363, 52369, 52543, 52583, 52769, 52783, 52903, 52967, 52981, 54101, 54167, 54323, 54503, 54521, 54541, 54547, 54563, 54583, 54727, 54787, 54949, 54983, 56101, 56383, 56503, 56509, 56543, 56747, 56783, 56923, 56929, 56963, 58321, 58369, 58549, 58741, 58763, 58789, 58901, 58909, 58921, 58943, 58967, 70123, 70141, 70163, 70183, 70327, 70381, 70507, 70549, 70589, 70729, 70901, 70921, 70981, 72103, 72109, 72169, 72341, 72383.

Some larger examples :