Metadromes

To quote from Numbers Aplenty:

A number is a metadrome in a given base \(b\) (often 10 or 16) if its digits are in strictly increasing order in that base. For example, 1234, 68 and 12789 are all metadromes in base 10. The total number of metadromes in base \(b\)  is equal to \(2^{b-1}\), hence in base 10 there are \(2^{10-1} = 2^9=512\) metadromes ranging from 0 to 123456789.  

For some reason, I've ignored these numbers over the years even though they make their appearance regularly in Numbers Aplenty. Here is the full list of base 10 metadromes:

1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 13, 14, 15, 16, 17, 18, 19, 23, 24, 25, 26, 27, 28, 29, 34, 35, 36, 37, 38, 39, 45, 46, 47, 48, 49, 56, 57, 58, 59, 67, 68, 69, 78, 79, 89, 123, 124, 125, 126, 127, 128, 129, 134, 135, 136, 137, 138, 139, 145, 146, 147, 148, 149, 156, 157, 158, 159, 167, 168, 169, 178, 179, 189, 234, 235, 236, 237, 238, 239, 245, 246, 247, 248, 249, 256, 257, 258, 259, 267, 268, 269, 278, 279, 289, 345, 346, 347, 348, 349, 356, 357, 358, 359, 367, 368, 369, 378, 379, 389, 456, 457, 458, 459, 467, 468, 469, 478, 479, 489, 567, 568, 569, 578, 579, 589, 678, 679, 689, 789, 1234, 1235, 1236, 1237, 1238, 1239, 1245, 1246, 1247, 1248, 1249, 1256, 1257, 1258, 1259, 1267, 1268, 1269, 1278, 1279, 1289, 1345, 1346, 1347, 1348, 1349, 1356, 1357, 1358, 1359, 1367, 1368, 1369, 1378, 1379, 1389, 1456, 1457, 1458, 1459, 1467, 1468, 1469, 1478, 1479, 1489, 1567, 1568, 1569, 1578, 1579, 1589, 1678, 1679, 1689, 1789, 2345, 2346, 2347, 2348, 2349, 2356, 2357, 2358, 2359, 2367, 2368, 2369, 2378, 2379, 2389, 2456, 2457, 2458, 2459, 2467, 2468, 2469, 2478, 2479, 2489, 2567, 2568, 2569, 2578, 2579, 2589, 2678, 2679, 2689, 2789, 3456, 3457, 3458, 3459, 3467, 3468, 3469, 3478, 3479, 3489, 3567, 3568, 3569, 3578, 3579, 3589, 3678, 3679, 3689, 3789, 4567, 4568, 4569, 4578, 4579, 4589, 4678, 4679, 4689, 4789, 5678, 5679, 5689, 5789, 6789, 12345, 12346, 12347, 12348, 12349, 12356, 12357, 12358, 12359, 12367, 12368, 12369, 12378, 12379, 12389, 12456, 12457, 12458, 12459, 12467, 12468, 12469, 12478, 12479, 12489, 12567, 12568, 12569, 12578, 12579, 12589, 12678, 12679, 12689, 12789, 13456, 13457, 13458, 13459, 13467, 13468, 13469, 13478, 13479, 13489, 13567, 13568, 13569, 13578, 13579, 13589, 13678, 13679, 13689, 13789, 14567, 14568, 14569, 14578, 14579, 14589, 14678, 14679, 14689, 14789, 15678, 15679, 15689, 15789, 16789, 23456, 23457, 23458, 23459, 23467, 23468, 23469, 23478, 23479, 23489, 23567, 23568, 23569, 23578, 23579, 23589, 23678, 23679, 23689, 23789, 24567, 24568, 24569, 24578, 24579, 24589, 24678, 24679, 24689, 24789, 25678, 25679, 25689, 25789, 26789, 34567, 34568, 34569, 34578, 34579, 34589, 34678, 34679, 34689, 34789, 35678, 35679, 35689, 35789, 36789, 45678, 45679, 45689, 45789, 46789, 56789, 123456, 123457, 123458, 123459, 123467, 123468, 123469, 123478, 123479, 123489, 123567, 123568, 123569, 123578, 123579, 123589, 123678, 123679, 123689, 123789, 124567, 124568, 124569, 124578, 124579, 124589, 124678, 124679, 124689, 124789, 125678, 125679, 125689, 125789, 126789, 134567, 134568, 134569, 134578, 134579, 134589, 134678, 134679, 134689, 134789, 135678, 135679, 135689, 135789, 136789, 145678, 145679, 145689, 145789, 146789, 156789, 234567, 234568, 234569, 234578, 234579, 234589, 234678, 234679, 234689, 234789, 235678, 235679, 235689, 235789, 236789, 245678, 245679, 245689, 245789, 246789, 256789, 345678, 345679, 345689, 345789, 346789, 356789, 456789, 1234567, 1234568, 1234569, 1234578, 1234579, 1234589, 1234678, 1234679, 1234689, 1234789, 1235678, 1235679, 1235689, 1235789, 1236789, 1245678, 1245679, 1245689, 1245789, 1246789, 1256789, 1345678, 1345679, 1345689, 1345789, 1346789, 1356789, 1456789, 2345678, 2345679, 2345689, 2345789, 2346789, 2356789, 2456789, 3456789, 12345678, 12345679, 12345689, 12345789, 12346789, 12356789, 12456789, 13456789, 23456789, 123456789 

Of these 512 metadromes, 100 or about 20% are prime. \(p_{13479}=145679\) is the largest metadromic prime whose index is a metadrome too. Here are the primes:

2, 3, 5, 7, 13, 17, 19, 23, 29, 37, 47, 59, 67, 79, 89, 127, 137, 139, 149, 157, 167, 179, 239, 257, 269, 347, 349, 359, 367, 379, 389, 457, 467, 479, 569, 1237, 1249, 1259, 1279, 1289, 1367, 1459, 1489, 1567, 1579, 1789, 2347, 2357, 2389, 2459, 2467, 2579, 2689, 2789, 3457, 3467, 3469, 4567, 4679, 4789, 5689, 12347, 12379, 12457, 12479, 12569, 12589, 12689, 13457, 13469, 13567, 13679, 13789, 15679, 23459, 23567, 23689, 23789, 25679, 34589, 34679, 123457, 123479, 124567, 124679, 125789, 134789, 145679, 234589, 235679, 235789, 245789, 345679, 345689, 1234789, 1235789, 1245689, 1456789, 12356789, 23456789

Of these 512 metadromes, 131 have prime factors that are also metadromes. The largest of these is \(1235689 = 7 \times 13 \times 37 \times 367\). Here are the numbers:

1, 4, 6, 8, 9, 12, 14, 15, 16, 18, 24, 25, 26, 27, 28, 34, 35, 36, 38, 39, 45, 46, 48, 49, 56, 57, 58, 68, 69, 78, 125, 126, 128, 134, 135, 136, 138, 145, 147, 148, 156, 158, 168, 169, 178, 189, 234, 235, 236, 237, 238, 245, 247, 256, 259, 267, 268, 278, 289, 345, 348, 356, 357, 358, 368, 378, 456, 459, 468, 469, 478, 567, 578, 1235, 1239, 1246, 1248, 1256, 1258, 1269, 1345, 1357, 1368, 1369, 1456, 1458, 1468, 1479, 1568, 2345, 2346, 2349, 2368, 2457, 2478, 2479, 2569, 2578, 2679, 3456, 3458, 3468, 3478, 3578, 5678, 12348, 12358, 12467, 12478, 12789, 13456, 13467, 13468, 13579, 13689, 15678, 23569, 24589, 24678, 24679, 25678, 34568, 34569, 124579, 124689, 134568, 134589, 134689, 234567, 234689, 1235689

There are only 30 numbers that are metadromes in base 8 and base 10 and these are (permalink):

1, 2, 3, 4, 5, 6, 7, 12, 13, 14, 15, 19, 23, 28, 29, 37, 38, 39, 46, 47, 156, 157, 158, 159, 167, 238, 239, 247, 678, 679

See Table 1 for the conversions to base 8. There are of course only \(2^{8-1} = 2^7 =128 \) metadromes in base 8. 

Table 1: permalink

The algorithm that the permalink refers to can be used to generate similar tables for bases 9, 7, 6, 5, 4, 3 and 2, although for 2 there is only the number 1 that satisfies. The algorithm will not work for bases greater than 10, although it would be interesting to try to develop one.