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apocalyptic n.  and  twin primes

First numbers that belong to both sets:

823, 857, 859, 1019, 1031, 1277, 1279, 1289, 1301, 1303, 1321, 1429, 1483, 1487, 1667, 1669, 1723, 1871, 1933, 2081, 2083, 2113, 2131, 2143, 2239, 2267, 2269, 2311, 2383, 2549, 2551, 2591, 2593, 2657, 2659, 2687, 2789, 2791, 2801, 2971, 3001, 3119, 3169, 3251, 3253, 3257, 3329, 3331, 3371, 3373, 3389, 3391, 3461, 3463, 3467, 3469, 3557, 3559, 3581, 3583, 3671, 3673, 3767, 3821, 3851, 3853, 3919, 3929, 4021, 4049, 4051, 4091, 4093, 4157, 4159, 4217, 4219, 4229, 4231, 4261, 4271, 4273, 4339, 4421, 4423, 4517, 4519, 4547, 4549, 4637, 4649, 4651, 4787, 4789, 4799, 4801, 4931, 4933, 4967, 4969, 5009, 5021, 5231, 5233, 5417, 5419, 5441, 5443, 5477, 5479, 5501, 5503, 5519, 5521, 5639, 5641, 5651, 5653, 5657, 5659, 5741, 5743, 5849, 5851, 5867, 5879, 5881, 6091, 6131, 6133, 6197, 6199, 6269, 6271, 6299, 6301, 6359, 6361, 6449, 6451, 6551, 6553, 6569, 6571, 6659, 6661, 6691, 6701, 6703, 6761, 6779, 6781, 6793, 6829, 6869, 6871, 6947, 6949, 6959, 6961, 7127, 7129, 7211, 7213, 7307, 7309, 7331, 7333, 7349, 7351, 7457, 7459, 7489, 7547, 7559, 7561, 7589, 7591, 7757, 7759, 7877, 7879, 7949, 7951, 8009, 8011, 8087, 8089, 8219, 8221, 8231, 8233, 8291, 8293, 8387, 8389, 8429, 8431, 8537, 8539, 8597, 8599, 8627, 8629, 8819, 8821, 8837, 8839, 8861, 8863, 8969, 8971, 8999, 9001, 9011, 9013, 9041, 9043, 9239, 9241, 9281, 9283, 9341, 9343, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9629, 9677, 9679, 9719, 9721, 9767, 9769, 9857, 9859, 9929, 9931, 10007, 10009, 10037, 10039, 10067, 10069, 10091, 10093, 10139, 10141, 10271, 10273, 10301, 10303, 10331, 10333, 10427, 10429, 10457, 10459, 10499, 10501, 10529, 10531, 10709, 10711, 10859, 10861, 10889, 10891, 10937, 10939, 11057, 11071, 11117, 11119, 11159.

Some larger examples :