Xenodromes

Ben 10 Ultimate Alien: Xenodrome is a fighting
mobile game that was released on mobile devices.

This post has nothing to do with the above mentioned game instead I noticed that the number associated with my diurnal age today, 27594, has no repeating digits and I wondered if there was a word to describe such a number. Well, the OEIS uses the term "xenodrome" and, in base 10, there are 8,877,691 of them with the first being 0 and the last being 9,876,543,210. These numbers form OEIS A010784. Numbers of this sort are not listed in Numbers Aplenty. In the range up to 40,000, there are 14,346 xenodromes.

With so many numbers, it's best to apply some sort of sieve and one that comes to be mind is the metadrome, a number in which the digits are in strictly increasing order. If we look at numbers that are both xenodromes and metadromes in the range up to 40,000, we find that there are only 375 of them:

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

If we were to consider katadromes instead of metadromes, then in the range up to 40,000 the highest number can only be 9876 so that's a little too restrictive. Katadromes are numbers in which the digits are in strictly decreasing order. See blog posts Metadromes and Katadromes.

Of course numbers can be xenodromes in other bases and 27594 serves as a good example because not only is it a xenodrome in base 10 but also in bases 8, 9, 11 and 12 and others as well no doubt:$$ \begin{align} 27594_{10} &= 41760_{\, 9} \\&= 65712_{\, 8} \\&= 19806_{11} \\&= 13b76_{12} \end{align}$$

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.