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prime n.  and  self-describing n.

First numbers that belong to both sets:

10153331, 10173133, 10233221, 10311533, 10322321, 12103331, 12163133, 12163331, 12193133, 12311933, 12313319, 15103133, 15233221, 15311633, 15331931, 15333119, 16153133, 16153331, 16173133, 16331531, 16331831, 16333117, 17143331, 17311633, 17331031, 18103133, 18103331, 18123331, 18153133, 18163331, 18233221, 18322123, 18331031, 18331231, 18331531, 19123133, 19123331, 19143133, 19143331, 19311833, 19331831, 21183223, 21193223, 21321523, 21321623, 21322319, 23183221, 23213219, 23244223, 23321521, 23321621, 23422423, 29422429, 31101533, 31121633, 31121833, 31123319, 31143317, 31161433, 31163317, 31181533, 31183319, 31331617, 32102321, 32162321, 32211623, 32212133, 32212319, 32231621, 32242433, 32272733, 33103117, 33121931, 33141931, 33151231, 33151931, 33161231, 33161831, 33181231, 33311017, 33311417, 33311519, 33322727, 1017223133, 1017223331, 1019312233, 1019332231, 1022163133, 1022311733, 1022311933, 1022331631, 1031162233, 1031223317, 1031332219, 1033223119, 1041134441, 1041154441, 1041164441, 1041441941, 1041444113, 1044411241, 1044411941, 1044414113, 1210444141, 1213414441, 1244134141, 1244411341, 1244411741, 1244414117, 1316444141, 1319414441, 1341104441, 1341194441, 1341414419, 1341441841, 1344124141, 1344164141, 1344194141, 1344411941, 1415223133, 1415332231, 1416332231, 1422153133, 1422153331, 1422163331, 1422331931, 1431221633, 1433162231, 1433221631, 1433221831, 1433312219, 1510414441, 1517223133, 1518312233, 1522331431, 1531142233, 1531221733, 1533221831, 1533223117, 1541104441, 1541174441, 1541194441, 1541414419, 1544411741, 1610223133, 1613444141, 1615414441, 1617312233, 1633223117, 1633312217, 1641414419, 1641441541, 1644414113, 1710332231, 1712414441, 1715223331, 1715312233, 1716223331, 1722153133, 1722311033, 1722311933, 1722313319, 1733221031, 1733221531, 1744411541, 1813444141, 1814223133, 1816414441, 1817444141, 1822143331, 1822153133, 1831221533, 1833221431, 1833221531, 1841444119, 1844174141, 1844194141, 1844414119, 1910223331, 1910312233, 1910332231, 1910444141, 1913414441, 1914223133, 1914223331, 1914312233, 1922331431, 1922333117, 1931172233, 1931221433, 1933221431, 1941124441, 1941154441, 1941164441, 1941414413, 1941441041, 1941441241, 2142424421, 2142444221, 2210331631, 2210333117, 2210333119, 2214183331, 2214311833, 2214331531, 2214331631, 2215183331, 2215311433, 2215311733, 2216311033, 2216313317, 2216333117, 2217163133, 2217163331, 2218311533, 2219173133, 2219311433, 2219331431, 2231103319, 2231171933, 2231191733, 2231331019, 2231331917, 2233101931, 2233141531, 2233143119, 2233173119, 2233311419, 2233311617, 2742424427, 3110192233, 3110221633, 3114182233, 3114192233, 3114332219, 3116102233, 3117221033, 3117221633, 3117332219, 3119221433, 3122101733, 3122141933, 3122173319, 3133142219, 3133162217, 3133192217, 3310221731, 3310223117, 3314221931, 3315182231, 3316221431, 3316223117, 3316312217, 3317221031, 3317221631, 3317221931, 3318142231, 3322103117, 3322173119, 3322311017, 3322311617, 3331102219, 3331192217, 3331221917, 4110194441, 4110441241, 4112414419, 4113104441, 4116104441, 4116441541, 4116444119, 4118134441, 4118174441, 4118194441, 4118414413, 4119154441, 4119441341, 4141124413, 4141134419, 4144104119.

Some larger examples :