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hoax n.  and  unprimeable n.

First numbers that belong to both sets:

516, 535, 895, 1148, 1344, 1642, 1678, 1795, 1858, 1894, 1924, 1966, 2106, 2182, 2325, 2326, 2362, 2515, 2565, 2826, 3172, 3246, 3268, 3285, 3474, 3476, 3484, 3505, 3892, 4148, 4160, 4162, 4188, 4198, 4306, 4414, 4744, 4765, 4770, 4776, 4820, 4824, 4844, 4855, 4920, 5062, 5068, 5242, 5422, 5425, 5458, 5464, 5548, 5602, 5612, 5674, 5752, 5946, 5998, 6184, 6500, 6502, 6504, 6512, 6620, 6934, 7268, 7402, 7428, 7465, 7650, 7762, 7834, 7978, 8076, 8158, 8185, 8308, 8334, 8720, 8724, 8790, 8914, 8982, 9116, 9306, 9350, 9355, 9382, 9598, 9650, 9652, 9772, 9778, 9895, 9916, 9985, 10044, 10375, 10406, 10675, 10806, 10926, 10966, 11124, 11228, 12055, 12068, 12188, 12318, 12366, 12388, 12442, 12464, 12622, 12664, 12775, 12932, 13194, 13408, 13506, 13588, 13748, 13770, 13815, 13974, 14046, 14092, 14268, 14276, 14396, 14458, 14494, 14520, 14644, 14998, 15024, 15115, 15150, 15205, 15500, 15596, 15712, 15835, 15898, 16015, 16285, 16294, 16500, 16582, 16684, 16735, 16770, 16850, 16860, 16890, 17086, 17145, 17148, 17175, 17212, 17225, 17226, 17276, 17364, 17455, 17524, 17616, 17635, 17646, 17698, 17754, 17812, 17964, 18022, 18025, 18032, 18274, 18278, 18474, 18562, 18564, 18624, 18650, 18844, 18922, 18985, 18994, 19118, 19170, 19175, 19176, 19348, 19498, 19516, 19524, 19564, 19615, 19678, 20376, 20516, 20578, 20728, 20970, 20974, 21478, 21625, 21694, 21955, 22086, 22234, 22266, 22352, 22475, 22522, 22550, 22558, 22585, 22590, 22608, 22828, 22950, 23238, 23242, 23650, 23658, 23694, 23725, 23790, 23932, 23962, 24142, 24214, 24216, 24260, 24264, 24272, 24304, 24306, 24345, 24560, 24585, 24640, 24668, 24754, 25000, 25006, 25105, 25132, 25195, 25275, 25294, 25298, 25484, 25508, 25512, 25618, 25834, 25850, 26060, 26365, 26545, 26580, 26914, 27085, 27188, 27355, 27382, 27418, 27516, 27562, 27598, 27600, 27604, 27660, 28132, 28174, 28252, 28255, 28322.

Some larger examples :