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emirpimes  and  repunit

First numbers that belong to both sets:

15, 85, 183, 341, 381, 511, 553, 781, 871, 1057, 1561, 1641, 1807, 1893, 1981, 2047, 2257, 4369, 7141, 7311, 8557, 9121, 9313, 10713, 11773, 15501, 16003, 16105, 16257, 19741, 30451, 32221, 33307, 36291, 38221, 39007, 39801, 43057, 50401, 54991, 66823, 70491, 75351, 76453, 78681, 89701, 91507, 92113, 97033, 100173, 101443, 102721, 103363, 104653, 109231, 111151, 115261, 119371, 133591, 141001, 141753, 150933, 151711, 152491, 161203, 167691, 169333, 170983, 175981, 176821, 183613, 187057, 194923, 196693, 199363, 204205, 228007, 260611, 266773, 300853, 311923, 313041, 315283, 318661, 325471, 336981, 346333, 352243, 357007, 381307, 396271, 411523, 434941, 456301, 475411, 476791, 497731, 501973, 549823, 557263, 566257, 593671, 597871, 602953, 623311, 666673, 689731, 711493, 723351, 730171, 738741, 761257, 773521, 780573, 794773, 819931, 892081, 926407, 930261, 934123, 973183, 1009021, 1057813, 1070191, 1080561, 1082401, 1095163, 1114081, 1126783, 1158853, 1173973, 1211101, 1239883, 1293907, 1300741, 1356061, 1363057, 1370071, 1381801, 1388863, 1412533, 1448413, 1467733, 1477441, 1479873, 1496953, 1504303, 1519057, 1526461, 1528933, 1533883, 1544761, 1556257, 1571263, 1586341, 1611631, 1632007, 1657657, 1662811, 1673143, 1686103, 1693903, 1717411, 1720033, 1733173, 1741081, 1754301, 1756951, 1764913, 1780891, 1783561, 1786233, 1788907, 1807681, 1810371, 1813063, 1823851, 1859133, 1872793, 1878271, 1936273, 1948717, 1969813, 1972621, 1975431, 1978243, 1986691, 1989511, 1995157, 2049193, 2054923, 2278591, 2293711, 2491663, 2603383, 2684683, 2803951, 2881507, 3018907, 3025861, 3078271, 3099361, 3109933, 3120523, 3131131, 3141757, 3152401, 3155953, 3188011, 3205891, 3238201, 3256221, 3259831, 3270673, 3296041, 3369061, 3376407, 3383761, 3397651, 3402181, 3446593, 3461461, 3468907, 3480091, 3495031, 3517501, 3525007, 3532521, 3547573, 3573991, 3585343, 3820909, 3839641, 3871057, 3958111, 3966073, 3982021, 3986013, 3990007, 4122931, 4320163, 4345141, 4382743, 4386931, 4500763, 4547557, 4793911, 4904011, 4917307, 4930621, 4979593, 5073757, 5600323, 5614531, 5858821, 5873353, 5951161, 5960923, 6202591, 6272521, 6438907, 6689983, 6825157, 6861781, 7046371, 7078261, 7131571, 7158301, 7244173, 7292701, 7325143, 7379373, 7384807, 7454981, 7499383, 7532281, 7570753, 7587271, 7598293, 7620361.

Some larger examples :