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equidigital n.  and  nialpdrome

First numbers that belong to both sets:

1, 2, 3, 5, 7, 10, 11, 21, 31, 32, 41, 43, 53, 61, 64, 71, 73, 81, 83, 97, 111, 211, 311, 320, 331, 421, 431, 433, 443, 511, 521, 541, 553, 631, 640, 641, 643, 653, 661, 733, 743, 751, 761, 773, 811, 821, 841, 853, 863, 875, 877, 881, 883, 887, 911, 941, 953, 961, 971, 977, 983, 991, 997, 1000, 2000, 2111, 2221, 3200, 3221, 3321, 3331, 4000, 4111, 4211, 4331, 4333, 4421, 4441, 5000, 5333, 5411, 5431, 5441, 5443, 5521, 5531, 6211, 6221, 6311, 6400, 6421, 6433, 6521, 6551, 6553, 6653, 6655, 6661, 7211, 7321, 7331, 7333, 7411, 7433, 7442, 7533, 7541, 7552, 7621, 7643, 7663, 7741, 7753, 7776, 8000, 8111, 8221, 8311, 8431, 8443, 8521, 8543, 8633, 8641, 8663, 8731, 8741, 8750, 8753, 8761, 8821, 8831, 8861, 8863, 8875, 8887, 9221, 9311, 9421, 9431, 9433, 9511, 9521, 9533, 9551, 9631, 9643, 9661, 9721, 9733, 9743, 9811, 9833, 9851, 9871, 9875, 9883, 9887, 9931, 9941, 9963, 9973, 11111, 22111, 22221, 30000, 31111, 32221, 33211, 33311, 33331, 41111, 42221, 43211, 43321, 43331, 44111, 44221, 44311, 44411, 44443, 51111, 52111, 53221, 53311, 53321, 54221, 54311, 54331, 54421, 54432, 54443, 55111, 55331, 55333, 55411, 55433, 55441, 55511, 55531, 55541, 55543, 55552, 55553, 60000, 63211, 63311, 63331, 64211, 64333, 64421, 64433, 65111, 65321, 65333, 65443, 65521, 65522, 65533, 65543, 65551, 66211, 66221, 66311, 66431, 66533, 66541, 66542, 66553, 66643, 66653, 66663, 70000, 72211, 72221, 73331, 74311, 74411, 74441, 75000, 75211, 75431, 75511, 75520, 75521, 75533, 75541, 75553, 76221, 76333, 76421, 76441, 76511, 76541, 76543, 76631, 76651, 77431, 77521, 77543, 77551, 77611, 77621, 77641, 77711, 77731, 77743, 77760, 77761, 77771, 77773, 83221, 83311, 84211, 84221, 84321, 84421, 84431, 84443, 85331.

Some larger examples :