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:
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:$$7210_{16} = 29200_{10} \\ 9876_{16}=39030_{10}$$
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