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balanced prime  and  centered square n.

First numbers that belong to both sets:

5, 15313, 20201, 156241, 353641, 1035361, 6340361, 6885761, 7507813, 8400901, 13204661, 15119501, 17588381, 20104141, 20872261, 22237781, 25283161, 25311613, 41724113, 44076661, 44189401, 49511201, 55598513, 55978781, 59721521, 65608513, 77314613, 82805581, 101887813, 108412813, 156096781, 157904221, 159471941, 190808113, 190886261, 201101513, 215177513, 240352813, 252023701, 258349181, 260353381, 261404113, 271654741, 273429113, 279306613, 290525513, 301572241, 356311513, 395957941, 422212741, 437044613, 501779521, 517808381, 519225313, 543675313, 546977813, 587182181, 603955013, 631581341, 641142241, 687055381, 710230361, 758979761, 874916281, 906145021, 967780013, 972801941, 978544561, 1030988641, 1066804241, 1079312261, 1100696281, 1107839521, 1142659013, 1194089581, 1332260581, 1339807613, 1343952013, 1374924361, 1428184013, 1430643541, 1440288121, 1480169641, 1501794013, 1516627813, 1552637813, 1587604901, 1631632813, 1715749621, 1726897681, 1740795013, 1779120901, 1877925613, 1883199821, 1909681801, 1925783861, 1973490313, 2007941821, 2038347401, 2060242241, 2160808061, 2222111113, 2280758261, 2381949221, 2438463613, 2605625861, 2735114761, 2765820313, 2768498461, 2792660113, 2812575001, 2832109061, 2853070261, 2886100313, 2896442161, 2967966013, 3036620381, 3045823201, 3067716121, 3111106081, 3150671581, 3283011481, 3307365781, 3344720261, 3418073881, 3509627981, 3589671181, 3610885181, 3749347013, 3903656441, 3978766013, 4041454513, 4077857741, 4147692121, 4197203821, 4242033941, 4285435621, 4442379541, 4447282361, 4570731661, 4632127501, 4726933681, 4777042513, 4777433501, 4784474021, 4831936513, 4843149781, 4870351513, 4946046341, 4975530013, 5020521013, 5111110513, 5176022513, 5197411013, 5270667121, 5289986941, 5340714601, 5607299101, 5753962813, 5829300313, 5886016501, 5982539113, 5986258781, 6157725313, 6224936621, 6260916901, 6296551981, 6391394861, 6399085321, 6448958881, 6602268961, 6752265841, 7401672781, 7484638901, 7637862013, 7924271941, 7949470141, 8058786013, 8078603161, 8115507401, 8186752841, 8343090313, 8476150201, 8549219561, 8604227381, 8741196421, 8834527813, 8844367001, 9286255481, 9293616113, 9321722341, 9494282201, 9620093341, 9992597081.