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happy n.  and  repunit

First numbers that belong to both sets:

7, 13, 31, 91, 133, 820, 931, 1093, 1333, 2257, 3193, 3616, 3661, 3907, 5257, 6163, 6643, 9313, 9331, 9703, 10303, 11131, 11773, 13111, 14281, 20881, 22621, 22651, 23257, 24181, 24807, 28393, 29524, 29757, 30103, 32581, 35157, 36673, 37060, 38221, 39007, 41821, 45583, 47743, 47989, 51303, 54523, 55461, 60763, 62251, 65281, 66307, 66430, 68907, 73171, 75351, 83811, 84973, 85557, 90301, 92721, 93196, 93331, 102081, 105951, 106603, 109893, 112561, 115261, 117307, 118681, 127807, 135057, 141001, 153273, 155631, 158007, 164431, 165243, 173473, 178507, 180201, 182757, 185331, 186193, 187057, 187923, 189661, 193161, 194923, 200257, 204205, 208393, 214833, 217623, 219493, 221371, 222313, 227053, 231843, 232807, 233773, 243543, 246513, 254080, 258573, 262143, 268843, 270921, 274053, 283557, 284623, 286761, 288907, 298663, 303051, 322057, 328903, 330051, 341641, 343983, 351057, 354621, 355813, 360601, 363007, 365421, 376383, 378841, 385021, 390001, 398793, 402234, 416671, 417963, 419257, 420553, 424453, 434941, 438907, 449571, 462540, 464443, 465807, 478173, 482331, 486507, 487903, 516243, 517681, 522007, 526351, 529257, 542433, 551307, 552793, 563251, 570781, 579195, 582933, 587523, 598303, 636007, 637603, 643539, 648831, 652057, 671581, 673221, 684757, 718257, 723351, 730171, 731881, 737191, 738741, 759513, 763003, 764751, 771763, 775281, 823557, 839973, 851007, 856551, 869557, 873291, 875161, 886423, 890193, 901551, 907257, 909163, 911071, 918723, 924483, 928333, 937993, 939931, 959421, 961381, 969241, 975157, 987043, 989031, 995007, 1003003, 1011031, 1017073, 1031241, 1061931, 1070191, 1078483, 1082401, 1105653, 1109863, 1111111, 1116193, 1154551, 1189191, 1202313, 1236600, 1242111, 1271257, 1298461, 1303023, 1342123, 1358391, 1363057, 1370071, 1393581, 1407783, 1419673, 1434007, 1462891, 1489621, 1494507, 1528933, 1583823, 1591383, 1609093, 1616713, 1642243, 1680913, 1691301, 1706943, 1709557, 1742521, 1743721, 1764913, 1791583, 1794261, 1821151, 1830855, 1834671, 1850961, 1853683, 1859133, 1861861, 1870057, 1941843, 1958601, 1968876, 1972621, 1995157, 2000811, 2014981, 2023507, 2029201, 2049193, 2054923, 2083693, 2095257, 2101051, 2106853, 2112663, 2121393, 2130141, 2135983, 2138907.

Some larger examples :