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Honaker prime  and  strong prime

First numbers that belong to both sets:

457, 1049, 1091, 1301, 1361, 2141, 2591, 2707, 2729, 3433, 3631, 4091, 4153, 4357, 4903, 5009, 5711, 5741, 5801, 6311, 6353, 6563, 6653, 6827, 7013, 7283, 7411, 7741, 9103, 10559, 11251, 11821, 12277, 13007, 13217, 13327, 13367, 13417, 14447, 14843, 14923, 15761, 16451, 17159, 17189, 17317, 18041, 18307, 20533, 20543, 22189, 22277, 22613, 22807, 23531, 24019, 24091, 24733, 24841, 25219, 25229, 25339, 26309, 26339, 27059, 27091, 27277, 29101, 29243, 29423, 30253, 30293, 31177, 31391, 31567, 34259, 35149, 36307, 36433, 36523, 37019, 37039, 37049, 37117, 38711, 40427, 40627, 40637, 41131, 41141, 41333, 41801, 41941, 42323, 42373, 42737, 43781, 45119, 45191, 45821, 46723, 47147, 47293, 47491, 47623, 48109, 48119, 48463, 48473, 49031, 50047, 50119, 51131, 51473, 52051, 52553, 52691, 53407, 53437, 53591, 53881, 53923, 54617, 54941, 55049, 55541, 56807, 57089, 57139, 57943, 58411, 59183, 60251, 60601, 61051, 61253, 61703, 62273, 62323, 62617, 62627, 63607, 64151, 64231, 64621, 65371, 65713, 67073, 67801, 68813, 69257, 69463, 70039, 70139, 70379, 71261, 72019, 72337, 72701, 73009, 73607, 74713, 75503, 77237, 77339, 77611, 77711, 78163, 78427, 78437, 79133, 79273, 79333, 79609, 82207, 82217, 83231, 84313, 85081, 87071, 88661, 89833, 90163, 90469, 91009, 91571, 92347, 92861, 93097, 94033, 94151, 94219, 94327, 96043, 96053, 96443, 96553, 97103, 97151, 97231, 97711, 99041, 101869, 102359, 102397, 102829, 103979, 105031, 105331, 106031, 106903, 108037, 109133, 110119, 111491, 112153, 113327, 113357, 113417, 113891, 114451, 115601, 115631, 116089, 116159, 116189, 116257, 120181, 120473, 120709, 122471, 123701, 124193, 124363, 125117, 125383, 125423, 126143, 129113, 129221, 131009, 131213, 132283, 132541, 132607, 133153, 133241, 133271, 133801, 133811, 133831, 134731, 136177, 136207, 136217, 136237, 136531, 137437, 138319, 139831, 140143, 141301, 142231, 143443, 144307, 145063, 145207, 145451, 145543, 145721, 145771, 149173, 150287, 151337, 151597, 152003, 152851, 154211, 156127, 156253.

Some larger examples :