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:$$ \begin{align} 27594_{10} &= 41760_{\, 9} \\&= 65712_{\, 8} \\&= 19806_{11} \\&= 13b76_{12} \end{align}$$

Katadromes

Having dealt with metadromes in an earlier post, it's time to look at katadromes the opposite of these numbers. According to Numbers Aplenty (which site is now working again):

A number is a katadrome in a given base \(b\)  (often 10 or 16) if its digits are in strictly decreasing order in that base. For example, 43210, 76521 and 9630 are all katadromes in base 10. If we allow the digits of a katadrome to be non-strictly decreasing (i.e., nonincreasing, like in 43310 or 2222, we obtain nialpdromes. Similarly, the numbers whose digits are nondecreasing and strictly increasing are called plaindromes and metadromes, respectively. The total number katadromes in base \(b\)  is equal to \(2^b-1\), hence in base 10 there are \(2^{10}-1 = 1034 - 1 =1023\) katadromes, from 0 to 9876543210.

Here are the katadromes between 9000 and 90000 (link):

..., 9210, 9310, 9320, 9321, 9410, 9420, 9421, 9430, 9431, 9432, 9510, 9520, 9521, 9530, 9531, 9532, 9540, 9541, 9542, 9543, 9610, 9620, 9621, 9630, 9631, 9632, 9640, 9641, 9642, 9643, 9650, 9651, 9652, 9653, 9654, 9710, 9720, 9721, 9730, 9731, 9732, 9740, 9741, 9742, 9743, 9750, 9751, 9752, 9753, 9754, 9760, 9761, 9762, 9763, 9764, 9765, 9810, 9820, 9821, 9830, 9831, 9832, 9840, 9841, 9842, 9843, 9850, 9851, 9852, 9853, 9854, 9860, 9861, 9862, 9863, 9864, 9865, 9870, 9871, 9872, 9873, 9874, 9875, 9876, 43210, 53210, 54210, 54310, 54320, 54321, 63210, 64210, 64310, 64320, 64321, 65210, 65310, 65320, 65321, 65410, 65420, 65421, 65430, 65431, 65432, 73210, 74210, 74310, 74320, 74321, 75210, 75310, 75320, 75321, 75410, 75420, 75421, 75430, 75431, 75432, 76210, 76310, 76320, 76321, 76410, 76420, 76421, 76430, 76431, 76432, 76510, 76520, 76521, 76530, 76531, 76532, 76540, 76541, 76542, 76543, 83210, 84210, 84310, 84320, 84321, 85210, 85310, 85320, 85321, 85410, 85420, 85421, 85430, 85431, 85432, 86210, 86310, 86320, 86321, 86410, 86420, 86421, 86430, 86431, 86432, 86510, 86520, 86521, 86530, 86531, 86532, 86540, 86541, 86542, 86543, 87210, 87310, 87320, 87321, 87410, 87420, 87421, 87430, 87431, 87432, 87510, 87520, 87521, 87530, 87531, 87532, 87540, 87541, 87542, 87543, 87610, 87620, 87621, 87630, 87631, 87632, 87640, 87641, 87642, 87643, 87650, 87651, 87652, 87653, 87654, ...

In terms of my diurnal age it can be seen that 9876 was the last number associated with that and then there is the huge jump to 43210 that I'll never get to experience. An interesting fact is that :$$p_{8510}=87641$$is the largest katadromic prime whose index is a katadromic too.

However,  if we consider katadromes in base 16, then the decimal equivalents of these katadromes form OEIS A023797:

 A023797: Katadromes: digits in base 16 are in strict descending order.

Here are the decimal equivalents from 29000 to 40000 (permalink to Python code):

..., 29200, 29456, 29472, 29473, 29712, 29728, 29729, 29744, 29745, 29746, 29968, 29984, 29985, 30000, 30001, 30002, 30016, 30017, 30018, 30019, 30224, 30240, 30241, 30256, 30257, 30258, 30272, 30273, 30274, 30275, 30288, 30289, 30290, 30291, 30292, 33296, 33552, 33568, 33569, 33808, 33824, 33825, 33840, 33841, 33842, 34064, 34080, 34081, 34096, 34097, 34098, 34112, 34113, 34114, 34115, 34320, 34336, 34337, 34352, 34353, 34354, 34368, 34369, 34370, 34371, 34384, 34385, 34386, 34387, 34388, 34576, 34592, 34593, 34608, 34609, 34610, 34624, 34625, 34626, 34627, 34640, 34641, 34642, 34643, 34644, 34656, 34657, 34658, 34659, 34660, 34661, 37392, 37648, 37664, 37665, 37904, 37920, 37921, 37936, 37937, 37938, 38160, 38176, 38177, 38192, 38193, 38194, 38208, 38209, 38210, 38211, 38416, 38432, 38433, 38448, 38449, 38450, 38464, 38465, 38466, 38467, 38480, 38481, 38482, 38483, 38484, 38672, 38688, 38689, 38704, 38705, 38706, 38720, 38721, 38722, 38723, 38736, 38737, 38738, 38739, 38740, 38752, 38753, 38754, 38755, 38756, 38757, 38928, 38944, 38945, 38960, 38961, 38962, 38976, 38977, 38978, 38979, 38992, 38993, 38994, 38995, 38996, 39008, 39009, 39010, 39011, 39012, 39013, 39024, 39025, 39026, 39027, 39028, 39029, 39030, ...

As can be seen, given my current diurnal age of 27576 days, it will some time before I enjoy a katadromic day in base 16 (29200 - 27576 = 1624 days away corresponding to 14th March 2029). I have to confess to not understanding the Python code used to generate these numbers. Here are the hexadecimal katadromes from 29000 (decimal) to 40000 (decimal) - permalink:

7210, 7310, 7320, 7321, 7410, 7420, 7421, 7430, 7431, 7432, 7510, 7520, 7521, 7530, 7531, 7532, 7540, 7541, 7542, 7543, 7610, 7620, 7621, 7630, 7631, 7632, 7640, 7641, 7642, 7643, 7650, 7651, 7652, 7653, 7654, 8210, 8310, 8320, 8321, 8410, 8420, 8421, 8430, 8431, 8432, 8510, 8520, 8521, 8530, 8531, 8532, 8540, 8541, 8542, 8543, 8610, 8620, 8621, 8630, 8631, 8632, 8640, 8641, 8642, 8643, 8650, 8651, 8652, 8653, 8654, 8710, 8720, 8721, 8730, 8731, 8732, 8740, 8741, 8742, 8743, 8750, 8751, 8752, 8753, 8754, 8760, 8761, 8762, 8763, 8764, 8765, 9210, 9310, 9320, 9321, 9410, 9420, 9421, 9430, 9431, 9432, 9510, 9520, 9521, 9530, 9531, 9532, 9540, 9541, 9542, 9543, 9610, 9620, 9621, 9630, 9631, 9632, 9640, 9641, 9642, 9643, 9650, 9651, 9652, 9653, 9654, 9710, 9720, 9721, 9730, 9731, 9732, 9740, 9741, 9742, 9743, 9750, 9751, 9752, 9753, 9754, 9760, 9761, 9762, 9763, 9764, 9765, 9810, 9820, 9821, 9830, 9831, 9832, 9840, 9841, 9842, 9843, 9850, 9851, 9852, 9853, 9854, 9860, 9861, 9862, 9863, 9864, 9865, 9870, 9871, 9872, 9873, 9874, 9875, 9876

Notice how the hexadecimal digits A, B, C, D, E and F are not required to represent the decimal equivalents. As examples we can see that:$$7210_{16} = 29200_{10} \\ 9876_{16}=39030_{10}$$