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junction n.  and  repunit

First numbers that belong to both sets:

111, 307, 507, 511, 820, 1407, 3423, 6321, 8931, 9724, 11111, 12211, 12720, 13111, 14425, 17031, 19531, 20023, 20307, 28731, 29524, 32221, 34411, 38221, 41007, 41413, 41821, 44311, 52213, 55933, 66430, 69433, 71023, 72631, 86731, 93331, 97033, 104007, 134323, 136531, 144021, 155631, 164431, 168421, 169333, 185331, 194923, 198535, 222313, 238633, 244531, 255531, 265720, 281431, 287833, 335821, 349525, 353431, 363024, 366631, 394420, 411523, 482331, 497731, 510511, 533631, 592131, 599845, 607621, 632821, 647221, 699733, 737023, 738741, 756031, 798343, 819931, 830833, 837931, 865831, 877033, 884541, 934123, 987043, 991021, 1009021, 1021111, 1061931, 1111111, 1143831, 1197931, 1233211, 1278031, 1284823, 1291633, 1342123, 1353733, 1360723, 1376831, 1395943, 1426831, 1453231, 1467733, 1528933, 1538841, 1596433, 1604023, 1611631, 1649941, 1704331, 1720033, 1742521, 1770231, 1845523, 1903021, 1916841, 1922125, 1978243, 2000719, 2046331, 2159431, 2233531, 2266531, 2342431, 2351623, 2360833, 2374321, 2444533, 2453923, 2478736, 2542431, 2577631, 2596933, 2609841, 2763741, 2820721, 2837541, 2851033, 2871331, 2905321, 3005023, 3092323, 3095841, 3120523, 3257437, 3259831, 3281533, 3296041, 3314221, 3387441, 3391123, 3495031, 3627121, 3630931, 3653833, 3669141, 3676135, 3750033, 3839641, 3938241, 3969435, 3978031, 4010007, 4014013, 4018021, 4022031, 4042111, 4045120, 4147333, 4258033, 4270423, 4277935, 4433131, 4458433, 4488043, 4539031, 4551823, 4564633, 4581741, 4586023, 4680733, 4758943, 4772041, 4789533, 4908441, 5029624, 5150631, 5218941, 5229043, 5237233, 5315331, 5319943, 5380840, 5399043, 5431231, 5459233, 5562523, 5567241, 5576740, 5586133, 5614531, 5662021, 5704933, 5849143, 6083623, 6098431, 6157843, 6222531, 6277531, 6327741, 6366036, 6403431, 6469639, 6499951, 6602331, 6731431, 6793843, 6819933, 7072941, 7110223, 7228033, 7254943, 7319731, 7352233, 7373941, 7401121, 7455631, 7488433, 7636933, 7670131, 7697851, 7775733, 7904533, 8011731, 8171023, 8176741, 8199633, 8234031, 8378131, 8406840, 8476833, 8517643, 8605423, 8623033, 8646541, 8699551, 8800123, 8961043, 9069133, 9123421, 9244641, 9250723, 9385033, 9495643, 9538833, 9706341, 9768751, 9800031, 9837633, 10045731, 10109221, 10166533, 10172526, 10204831, 10211221, 10265620, 10275231, 10436131, 10474933, 10617823, 10755121, 10842924, 10926331, 10966033, 11092231, 11139520, 11158941, 11421021, 11522631, 11597431.

Some larger examples :