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happy n.  and  palindromic n.

First numbers that belong to both sets:

1, 7, 44, 262, 313, 383, 404, 464, 565, 656, 818, 888, 989, 1221, 1771, 1881, 2112, 3553, 4004, 4554, 4774, 5335, 5445, 5555, 7117, 7447, 7887, 8118, 8778, 11311, 11811, 12021, 12321, 12921, 14641, 15451, 15951, 17071, 17371, 18081, 18381, 20602, 21012, 21312, 21912, 22422, 22922, 24242, 24342, 25652, 25952, 26762, 29792, 30103, 30803, 33833, 34643, 35053, 36163, 36263, 36763, 40004, 40604, 41614, 42224, 42324, 43634, 44244, 44644, 45054, 45254, 45354, 46564, 47074, 47374, 48484, 49394, 49594, 49694, 50605, 51415, 51915, 52625, 52925, 53035, 54045, 54245, 54345, 55055, 55355, 56965, 58585, 59995, 60506, 62726, 63136, 63236, 63736, 64546, 65956, 66766, 66866, 67776, 68686, 71017, 71317, 74047, 74347, 76767, 78087, 78287, 78687, 80108, 80808, 81018, 81318, 84448, 85558, 86668, 87078, 87278, 87678, 88888, 89198, 89698, 90809, 92729, 94349, 94549, 94649, 95959, 98189, 98689, 102201, 107701, 108801, 120021, 123321, 126621, 132231, 135531, 153351, 162261, 169961, 170071, 179971, 180081, 196691, 197791, 199991, 201102, 210012, 213312, 216612, 231132, 233332, 239932, 256652, 261162, 265562, 268862, 286682, 293392, 305503, 312213, 315513, 321123, 323323, 329923, 332233, 334433, 335533, 343343, 344443, 345543, 350053, 351153, 353353, 354453, 359953, 367763, 376673, 392293, 395593, 400004, 405504, 407704, 433334, 434434, 435534, 443344, 448844, 449944, 450054, 453354, 466664, 470074, 484484, 494494, 499994, 503305, 504405, 505505, 513315, 526625, 530035, 531135, 533335, 534435, 539935, 540045, 543345, 550055, 559955, 562265, 579975, 593395, 595595, 597795, 612216, 619916, 621126, 625526, 628826, 637736, 646646, 652256, 664466, 673376, 682286, 689986, 691196, 698896, 701107, 704407, 708807, 710017, 719917, 736637, 740047, 759957, 763367, 780087, 791197, 795597, 801108, 807708, 810018, 826628, 844448, 862268, 869968, 870078, 896698, 916619, 917719, 919919, 923329, 932239, 935539, 944449, 949949, 953359, 955559, 957759, 961169, 968869, 971179, 975579, 986689, 991199, 994499, 1013101, 1018101, 1020201, 1023201, 1029201, 1046401, 1054501, 1059501.

Some larger examples :