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emirp  and  inconsummate n.

First numbers that belong to both sets:

761, 941, 971, 983, 1487, 1499, 1601, 1811, 3083, 3221, 3251, 3257, 3371, 3697, 3719, 3733, 3767, 3803, 3821, 7043, 7643, 7649, 9041, 9221, 9257, 9293, 9341, 9437, 9551, 9749, 10853, 11057, 11777, 11927, 12107, 12149, 12437, 12641, 13259, 13697, 13709, 14153, 14177, 14207, 14327, 14387, 14591, 14621, 14657, 14717, 14831, 14843, 14879, 14891, 14897, 14939, 15053, 15149, 15227, 15299, 15737, 16001, 16217, 16481, 16937, 17021, 17033, 17207, 17477, 17627, 17681, 17737, 18077, 18671, 19037, 19163, 19181, 19237, 19577, 19763, 19793, 30593, 30851, 30881, 30911, 31091, 31193, 31259, 31277, 31307, 31481, 31721, 31799, 31907, 31991, 32099, 32321, 32369, 32377, 32467, 32531, 32537, 32579, 32687, 32939, 33029, 33071, 33287, 33317, 33863, 33911, 34031, 34211, 34367, 34757, 34781, 34847, 35027, 35051, 35099, 35141, 35159, 35257, 35327, 35381, 35437, 35447, 35537, 35591, 36131, 36187, 36191, 36209, 36251, 36269, 36277, 36467, 36473, 36479, 36599, 36607, 36761, 36791, 36809, 36877, 36931, 37021, 37061, 37199, 37379, 37409, 37489, 37571, 37589, 37619, 37643, 37781, 37831, 38039, 38393, 38639, 38723, 38993, 39371, 39419, 39581, 39749, 39877, 39887, 70061, 70121, 70313, 70373, 70937, 71153, 71333, 71399, 71537, 72161, 72221, 72551, 73091, 73421, 73751, 74201, 75041, 75407, 75641, 76001, 76259, 77237, 77587, 77611, 79537, 90149, 90647, 90697, 90731, 90863, 90887, 90917, 91397, 91781, 92033, 92077, 92357, 92369, 92381, 92459, 92507, 92639, 92753, 93113, 93257, 93407, 93629, 93701, 93893, 94111, 94781, 94907, 95087, 95429, 95621, 96017, 96281, 96587, 96671, 96911, 97007, 97259, 97367, 97397, 97523, 98081, 98129, 98207, 98429, 98507, 98621, 98627, 98717, 98729, 99173, 99251, 99317, 99611, 100049, 100357, 100379, 100699, 101207, 101429, 101987, 102161, 102461, 102497, 102539, 102611, 102797, 103067, 103091, 103217, 103289, 103307, 103391, 103421, 103457, 103613, 103703, 103837, 106087, 106187, 106367, 106427, 106699, 106787, 107309, 108247, 108263.

Some larger examples :