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nialpdrome  and  powerful n.

First numbers that belong to both sets:

1, 4, 8, 9, 32, 64, 72, 81, 100, 200, 400, 432, 441, 500, 800, 841, 864, 900, 961, 972, 1000, 2000, 3200, 4000, 5000, 5400, 6400, 7200, 7744, 7776, 8000, 8100, 9000, 9800, 10000, 20000, 32000, 40000, 43200, 44100, 50000, 54000, 64000, 72000, 72200, 80000, 81000, 84100, 86400, 88200, 90000, 96100, 97200, 98000, 100000, 200000, 320000, 400000, 432000, 441000, 500000, 540000, 640000, 665500, 720000, 722000, 744200, 774400, 777600, 800000, 810000, 841000, 855432, 864000, 882000, 900000, 952200, 961000, 972000, 980000, 1000000, 2000000, 3200000, 4000000, 4320000, 4410000, 5000000, 5400000, 6400000, 6552200, 6655000, 7200000, 7220000, 7442000, 7744000, 7776000, 8000000, 8100000, 8410000, 8640000, 8666532, 8820000, 8874441, 9000000, 9522000, 9610000, 9720000, 9800000, 9853321, 9886500, 10000000, 20000000, 32000000, 40000000, 43200000, 44100000, 50000000, 54000000, 64000000, 65522000, 66550000, 72000000, 72200000, 74420000, 76632200, 77440000, 77760000, 80000000, 81000000, 84100000, 85543200, 86400000, 88200000, 90000000, 95220000, 96100000, 97200000, 98000000, 98865000, 100000000, 200000000, 320000000, 400000000, 432000000, 441000000, 500000000, 540000000, 640000000, 655220000, 665500000, 720000000, 722000000, 744200000, 766322000, 774400000, 777600000, 800000000, 810000000, 841000000, 855432000, 864000000, 866653200, 882000000, 887444100, 900000000, 952200000, 961000000, 972000000, 980000000, 985332100, 988650000, 999887641, 1000000000, 2000000000, 3200000000, 4000000000, 4320000000, 4410000000, 5000000000, 5400000000, 6400000000, 6552200000, 6655000000, 7200000000, 7220000000, 7442000000, 7663220000, 7744000000, 7776000000, 8000000000, 8100000000, 8410000000, 8554320000, 8640000000, 8666532000, 8820000000, 8874441000, 9000000000, 9522000000, 9610000000, 9720000000, 9800000000, 9853321000, 9886500000, 10000000000, 20000000000, 32000000000, 40000000000, 43200000000, 44100000000, 50000000000, 54000000000, 64000000000, 65522000000, 66550000000, 72000000000, 72200000000, 74420000000, 76632200000, 77440000000, 77760000000, 80000000000, 81000000000, 84100000000, 85543200000, 86400000000, 86665320000, 88200000000, 88744410000, 90000000000, 95220000000, 96100000000, 97200000000, 98000000000, 98533210000, 98865000000, 99988764100, 100000000000, 200000000000, 320000000000, 400000000000, 432000000000, 441000000000, 500000000000, 540000000000, 640000000000, 655220000000, 665500000000, 720000000000, 722000000000, 744200000000, 766322000000, 774400000000, 777600000000, 800000000000, 810000000000, 841000000000, 855432000000, 864000000000, 866653200000, 882000000000, 887444100000, 900000000000, 952200000000, 961000000000, 972000000000, 980000000000, 985332100000, 988650000000, 999887641000, 1000000000000, 2000000000000, 3200000000000, 4000000000000, 4320000000000, 4410000000000, 5000000000000, 5400000000000, 6400000000000, 6552200000000.

Some larger examples :