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hoax n.  and  powerful n.

First numbers that belong to both sets:

361, 1600, 2401, 2500, 16000, 25000, 33124, 39200, 39601, 81225, 102400, 130321, 135200, 143641, 160000, 161604, 171500, 207025, 250000, 283024, 308025, 323208, 392000, 406125, 480200, 546121, 612500, 731025, 780300, 1024000, 1095200, 1137032, 1147041, 1201216, 1352000, 1391112, 1540125, 1560600, 1600000, 1715000, 2030625, 2050624, 2080800, 2112500, 2223081, 2340900, 2500000, 2528100, 2592100, 3052009, 3240125, 3441123, 3493161, 3591200, 3901500, 3920000, 3996001, 4161600, 4263200, 4421700, 4512500, 4631104, 4802000, 4831204, 5056200, 5105212, 6022116, 6111008, 6125000, 6242400, 6245001, 6260004, 6502500, 7022700, 7128900, 7300804, 7527200, 7803000, 8323200, 8910225, 9210632, 10042561, 10153125, 10161032, 10240000, 10621081, 10952000, 11007432, 11704500, 12012500, 12044232, 12355225, 12440072, 12520008, 12640500, 12903200, 13265100, 13354112, 13520000, 13610700, 13764100, 14045400, 14257800, 14300552, 15103008, 15606000, 16000000, 16232841, 16601200, 17056900, 17112500, 17150000, 20025625, 20151121, 20224800, 20304036, 20421361, 20608200, 20711601, 20720704, 20808000, 21068100, 21125000, 21206025, 21255200, 21386700, 21735224, 22052416, 22108500, 22934521, 23112500, 23409000, 24452127, 24512401, 24700900, 25000000, 25040016, 25281000, 25921000, 26530200, 27221400, 28515600, 30195025, 30201992, 30206016, 30239001, 30337200, 30912300, 31047184, 31680800, 32004500, 32319225, 32364721, 32512500, 33200644, 33663204, 33756100, 34234201, 34445161, 34928100, 35113500, 35462025, 35644500, 35912000, 36048016, 36216324, 38812900, 39015000, 39200000, 40522400, 40832100, 41216400, 41616000, 42003361, 42016324, 42136200, 42390243, 42632000, 42773400, 44217000, 44551125, 44609041, 45024100, 45125000, 45630025, 47760300, 48020000, 50080032, 50353216, 50562000, 51344307, 51520500, 53060400, 53127432, 53363025, 53436100, 56085121, 56112500, 56310016, 60000516, 60152600, 60372900, 60412032, 61250000, 61383200, 62424000, 63202500, 63204300, 63303752, 63680400, 64071200, 64240225, 65025000, 65124900, 70227000, 70375321, 71289000, 72131049, 72216004, 75272000, 75516100, 78012500, 78030000, 79600500, 80192025, 81036004, 83232000, 84621601, 86510700, 90611361, 90630400, 91011600, 95520600.