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Duffinian n.  and  nude n.

First numbers that belong to both sets:

4, 8, 9, 36, 55, 77, 111, 115, 128, 144, 155, 175, 324, 333, 515, 777, 784, 999, 1111, 1115, 1197, 1296, 1444, 1555, 1575, 1771, 2888, 2916, 3171, 5155, 5515, 5555, 7777, 9216, 11111, 11133, 11155, 11515, 11555, 11664, 11711, 13131, 13797, 15115, 17717, 19719, 26244, 31113, 31131, 33399, 35175, 36864, 39375, 39393, 41616, 51115, 51155, 51515, 55115, 55555, 57155, 57575, 69696, 71337, 71771, 73731, 73773, 77133, 77931, 82944, 91917, 93339, 93393, 93636, 99333, 111115, 111515, 111755, 113133, 113193, 113319, 113331, 113913, 114244, 115555, 117775, 117999, 131133, 131331, 133113, 133131, 133191, 133311, 151115, 151555, 155555, 155575, 166464, 173775, 179991, 186624, 193113, 228488, 236196, 311133, 311319, 311331, 311913, 313173, 313311, 331311, 331317, 333111, 339399, 339993, 379575, 391311, 391797, 393399, 393993, 399339, 399393, 399933, 412164, 484416, 511115, 511175, 515155, 515555, 551155, 551515, 551555, 551775, 555115, 555155, 555515, 575155, 717115, 719775, 731997, 771155, 771715, 771939, 777175, 797391, 911331, 931131, 933399, 933993, 939933, 991179, 993933, 999117, 999333, 1111111, 1111113, 1111115, 1111555, 1111775, 1113111, 1113939, 1115155, 1115515, 1115555, 1117137, 1117711, 1119339, 1119933, 1131111, 1133919, 1139733, 1151515, 1151555, 1155115, 1155155, 1157555, 1171177, 1171317, 1177113, 1177771, 1193139, 1193319, 1199313, 1218816, 1241888, 1281424, 1311399, 1311939, 1311993, 1313316, 1313991, 1319193, 1319391, 1331337, 1331991, 1333773, 1339119, 1373337, 1377117, 1379133, 1382976, 1391319, 1391931, 1393911, 1399311, 1411344, 1468944, 1511155, 1511515, 1511555, 1515115, 1551115, 1551155, 1555555, 1577555, 1711171, 1711731, 1711773, 1717317, 1717555, 1719711, 1751155, 1771371, 1771777, 1775515, 1777111, 1777797, 1911933, 1913139, 1913931, 1931913, 1933911, 1973979, 1991313, 1993113, 1993311, 1997793, 2663424, 2862864, 3111111, 3111399, 3111696, 3111939, 3113397, 3113733, 3113919, 3119319, 3119913, 3131199, 3131793, 3139731, 3139911, 3191139, 3191193, 3191913, 3193191, 3196944, 3311199, 3317139, 3319119, 3331377, 3333339, 3339333, 3373713, 3393333, 3399111, 3663396, 3717777, 3771117, 3911931, 3931119, 3937311, 3973977, 3991113, 5111515, 5115515, 5151155, 5157775, 5377575, 5511115.

Some larger examples :