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balanced prime  and  Hogben n.

First numbers that belong to both sets:

157, 211, 1123, 3307, 5113, 30103, 35911, 77563, 628057, 731881, 830833, 2665057, 2710963, 2724151, 3472633, 4603171, 4654807, 5296903, 6966961, 7308913, 7770157, 8245513, 12684283, 13597657, 15354643, 16552693, 16838713, 18143341, 18220093, 19232611, 20761693, 25862311, 30930283, 33344851, 34745131, 36936007, 40685263, 46464673, 46751407, 47368807, 48490333, 51703291, 51789613, 54723007, 60614011, 62338921, 64008001, 64440757, 64826653, 66300307, 72428611, 93074257, 93789541, 102525751, 107920933, 109819921, 114436507, 114886243, 120615307, 126416293, 139700581, 148998643, 152263261, 154990051, 169846057, 181023571, 185926861, 200180053, 201341911, 204418507, 217931407, 221786557, 241103257, 260515741, 262650643, 264208771, 274018363, 298892233, 313945243, 328243807, 357228901, 362921551, 365211211, 371853373, 391545157, 411379807, 421173007, 453200233, 459866581, 469914007, 476614393, 484726273, 502230511, 508389757, 521779807, 546133531, 588717433, 620283931, 641431603, 667938181, 686518603, 716124361, 753694663, 787616161, 807611143, 869807557, 991462657, 1050505333, 1076332057, 1110722257, 1122752557, 1137679171, 1143690943, 1196641057, 1212885103, 1230922141, 1256312581, 1263980257, 1296180007, 1298989723, 1348101373, 1393640893, 1419970807, 1420875331, 1432054807, 1456376407, 1489381057, 1505401201, 1525019653, 1636566571, 1691389003, 1705731301, 1751213257, 1792802623, 1796615383, 1804847773, 1808333101, 1824784807, 1876405807, 1892554513, 1905802681, 1910258143, 1932437641, 1950856393, 1990096711, 2002070281, 2026035133, 2037123091, 2122030291, 2152449631, 2174590057, 2184674341, 2187666757, 2199469303, 2203692193, 2224018441, 2225810863, 2239135081, 2293691557, 2319241123, 2400951001, 2452676101, 2464576381, 2465370757, 2528732083, 2544950257, 2594221423, 2698334971, 2825613493, 2987442307, 3019557451, 3023185273, 3025275007, 3116263153, 3216534511, 3262751521, 3363246043, 3364870057, 3374622373, 3404314063, 3439177381, 3500556391, 3522600553, 3541142557, 3635667913, 3669027757, 3717279931, 3798811591, 3805964557, 3812877253, 3826721461, 4013285851, 4034345773, 4057498903, 4103491423, 4168058161, 4225975057, 4246933393, 4257888757, 4279841821, 4369804921, 4622844073, 4624612021, 4637541901, 4680543811, 4710557323, 4944410173, 4985560273, 5012144413, 5051015971, 5094033757, 5096746273, 5151291757, 5272575157, 5314337101, 5332723651, 5362120303, 5381469523, 5448728041, 5666551453, 5734351351, 5839022983, 5850337657, 5855080843, 5869628383, 5908766293, 5915225011, 5984337523, 6001523431, 6006017503, 6037212301, 6041875171, 6054940783, 6148833811, 6410644423, 6463114843, 6498858841, 6502890241, 6603431383, 6702123823, 6756922201, 6825155611, 6828625861, 6857247673, 6858903943, 6993558757, 7039797313, 7043993113, 7062133333, 7085178103, 7255887943, 7506056407, 7552218313, 7596255493, 7962439057, 8020366693, 8120983573, 8128194493, 8164477807, 8333955391, 8438718907, 8579298001, 8600614861, 8629573921, 8727389821, 8856033343, 8956445683, 8992634071, 8999083633, 9288622507, 9320068141, 9460577491, 9542066173, 9616057783, 9824675281, 9945973171.