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Duffinian n.  and  nialpdrome

First numbers that belong to both sets:

4, 8, 9, 21, 32, 50, 55, 63, 64, 65, 75, 77, 81, 85, 93, 98, 100, 111, 221, 333, 400, 511, 533, 543, 551, 553, 611, 633, 651, 655, 711, 721, 722, 731, 741, 755, 763, 777, 800, 831, 833, 841, 851, 865, 871, 873, 875, 900, 921, 943, 955, 961, 965, 975, 981, 985, 993, 999, 1111, 3211, 4221, 4321, 4331, 4333, 4411, 4431, 4433, 5111, 5211, 5221, 5311, 5321, 5331, 5411, 5421, 5533, 5543, 5555, 6331, 6400, 6411, 6431, 6433, 6443, 6511, 6533, 6541, 6543, 6611, 6631, 6641, 6651, 6655, 6663, 6665, 7111, 7311, 7421, 7442, 7511, 7531, 7533, 7543, 7555, 7631, 7633, 7641, 7651, 7653, 7655, 7661, 7663, 7711, 7721, 7731, 7744, 7751, 7761, 7763, 7765, 7771, 7775, 7777, 8100, 8321, 8333, 8411, 8433, 8531, 8533, 8551, 8553, 8611, 8621, 8633, 8643, 8651, 8661, 8665, 8711, 8743, 8751, 8755, 8765, 8771, 8773, 8777, 8833, 8841, 8843, 8851, 8855, 8873, 8875, 8881, 8885, 9111, 9211, 9331, 9443, 9541, 9543, 9553, 9611, 9621, 9641, 9651, 9653, 9655, 9665, 9731, 9751, 9755, 9761, 9763, 9773, 9775, 9777, 9821, 9841, 9853, 9863, 9865, 9881, 9911, 9921, 9943, 9953, 9955, 9961, 9965, 9971, 9977, 9983, 9985, 9991, 9993, 9997, 10000, 11111, 20000, 22221, 31111, 32111, 32221, 33221, 40000, 42211, 43111, 43221, 43311, 44211, 44311, 44321, 44411, 44431, 44441, 44443, 51111, 52111, 52211, 53111, 53211, 53221, 53311, 53321, 53333, 54211, 54221, 54333, 54431, 54433, 55111, 55211, 55321, 55421, 55431, 55433, 55443, 55521, 55531, 55543, 55551, 55553, 55555, 61111, 62111, 62221, 63111, 63221, 64111, 64211, 64321, 64331, 64411, 64421, 64431, 64441, 64443, 65311, 65321, 65333, 65411, 65431, 65441, 65443, 65522, 65531, 65533, 65555, 66211, 66311, 66321, 66331, 66333, 66421, 66433, 66441, 66443, 66511.

Some larger examples :