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twin primes  and  Ulam n.

First numbers that belong to both sets:

3, 11, 13, 197, 241, 431, 1489, 1721, 2789, 3371, 3373, 3527, 3671, 4517, 4519, 4969, 6569, 7459, 7559, 8011, 8539, 8969, 8971, 9013, 9721, 10271, 10457, 10459, 10501, 11779, 11969, 12917, 13399, 13721, 13931, 14083, 14251, 14449, 16069, 16651, 16829, 17389, 17909, 18251, 19427, 19429, 20441, 20443, 20807, 21017, 21191, 21193, 22483, 22637, 25169, 25171, 25469, 25601, 26701, 26879, 26891, 27109, 27241, 27541, 29401, 29569, 31123, 31729, 32057, 32059, 32189, 32191, 32411, 32941, 33589, 33809, 33811, 34261, 34757, 34759, 34847, 35729, 36013, 37547, 37811, 38669, 38671, 38749, 38923, 39509, 41413, 42407, 42901, 43783, 43961, 44699, 46441, 46681, 47419, 47741, 48479, 48821, 49031, 49171, 50549, 50551, 50891, 51199, 51971, 51973, 52709, 52711, 53549, 54629, 55621, 55817, 55819, 56503, 56533, 56921, 57527, 58153, 61561, 62297, 62299, 63421, 65581, 66749, 66751, 68713, 68821, 69151, 69493, 69857, 69859, 70573, 71413, 71711, 74099, 74101, 75709, 77549, 77551, 78509, 79841, 79843, 79997, 80207, 80447, 80449, 81551, 84317, 85199, 85331, 85817, 86533, 87181, 88801, 89561, 89563, 90401, 90619, 91459, 93253, 95801, 98909, 99259, 101531, 102499, 102677, 102929, 104231, 104383, 104551, 104761, 106543, 108461, 108463, 108959, 109199, 109201, 109453, 110731, 110921, 111119, 111271, 112067, 113023, 113147, 114797, 114799, 115769, 115933, 116189, 116927, 117191, 117883, 118247, 118249, 121577, 121787, 122611, 122743, 126493, 126611, 126613, 127033, 128201, 128939, 128969, 128971, 129223, 129289, 129401, 129587, 129589, 129917, 129919, 130369, 130843, 131449, 132047, 133279, 133543, 133711, 134681, 135463, 135979, 140731, 140839, 141511, 141767, 142591, 143111, 143263, 144167, 144169, 144409, 144541, 145601, 145721, 147211, 148861, 149159, 149161, 149249, 149911, 150991, 151903, 151969, 152027, 152531, 152533, 152897, 152939, 152941, 154669, 156677, 156941, 157427, 157559, 158143, 158747, 158749, 159631, 160091, 160093, 161339, 163411, 164447, 165703, 166739, 166849, 167023, 168599, 168601, 169891, 171541, 172423, 175081.

Some larger examples :