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

More On Digit Equations

I've posted before about rendering numbers as digit equations, specifically:

• Forming Equations from the Digits of a Number on the 14th March 2024
• The Number Plate Game on the 29th June 2024
• Forming Digit Equations: A Game on the 5th August 2024

Yesterday the number 27586 caught my attention because it can easily be rendered as a digit equation, viz.:$$27586 \rightarrow 2+7+5=8+6$$but it has the special quality that no digits are repeated. This got me wondering what other five digit numbers have this property. In the range of numbers from 27586 to 40000, there are 371 numbers that satisfy the two criteria:

• number has no repeating digits
• number can be split into two parts and a digit equation formed from the sum of the digits on either side of the equal sign

Here are the numbers (permalink):

27586, 27603, 27630, 27801, 27810, 28019, 28037, 28046, 28064, 28073, 28091, 28109, 28136, 28145, 28147, 28154, 28156, 28163, 28165, 28174, 28190, 28307, 28316, 28349, 28361, 28367, 28370, 28376, 28394, 28406, 28415, 28451, 28459, 28460, 28495, 28514, 28541, 28569, 28596, 28604, 28613, 28631, 28640, 28679, 28697, 28703, 28730, 28901, 28910, 29038, 29047, 29056, 29065, 29074, 29083, 29137, 29146, 29148, 29157, 29164, 29173, 29175, 29184, 29308, 29317, 29368, 29371, 29380, 29386, 29407, 29416, 29461, 29470, 29478, 29487, 29506, 29560, 29605, 29614, 29641, 29650, 29704, 29713, 29731, 29740, 29803, 29830, 30126, 30148, 30159, 30214, 30216, 30241, 30249, 30416, 30418, 30425, 30429, 30452, 30461, 30517, 30519, 30526, 30562, 30571, 30618, 30627, 30645, 30654, 30672, 30681, 30719, 30728, 30746, 30764, 30782, 30791, 30829, 30847, 30856, 30865, 30874, 30892, 30948, 30957, 30975, 30984, 31026, 31048, 31059, 31206, 31260, 31408, 31426, 31462, 31480, 31509, 31527, 31572, 31590, 31628, 31682, 31729, 31756, 31765, 31792, 31857, 31875, 31958, 31967, 31976, 31985, 32014, 32016, 32041, 32049, 32104, 32106, 32140, 32160, 32401, 32409, 32410, 32418, 32481, 32490, 32519, 32546, 32564, 32591, 32647, 32674, 32748, 32784, 32849, 32867, 32876, 32894, 32968, 32986, 34016, 34018, 34025, 34029, 34052, 34061, 34106, 34108, 34126, 34160, 34162, 34180, 34205, 34209, 34218, 34250, 34281, 34290, 34502, 34520, 34601, 34610, 34658, 34685, 34759, 34768, 34786, 34795, 34869, 34896, 35017, 35019, 35026, 35062, 35071, 35107, 35109, 35127, 35170, 35172, 35190, 35206, 35219, 35246, 35260, 35264, 35291, 35602, 35620, 35701, 35710, 35769, 35796, 35879, 35897, 36018, 36027, 36045, 36054, 36072, 36081, 36108, 36128, 36180, 36182, 36207, 36247, 36270, 36274, 36405, 36450, 36458, 36485, 36504, 36540, 36702, 36720, 36801, 36810, 37019, 37028, 37046, 37064, 37082, 37091, 37109, 37129, 37145, 37154, 37156, 37165, 37190, 37192, 37208, 37248, 37280, 37284, 37406, 37415, 37451, 37459, 37460, 37468, 37486, 37495, 37514, 37541, 37569, 37596, 37604, 37640, 37802, 37820, 37901, 37910, 38029, 38047, 38056, 38065, 38074, 38092, 38146, 38157, 38164, 38175, 38209, 38245, 38249, 38254, 38267, 38276, 38290, 38294, 38407, 38416, 38425, 38452, 38461, 38469, 38470, 38496, 38506, 38524, 38542, 38560, 38579, 38597, 38605, 38614, 38641, 38650, 38704, 38740, 38902, 38920, 39048, 39057, 39075, 39084, 39147, 39156, 39158, 39165, 39167, 39174, 39176, 39185, 39246, 39264, 39268, 39286, 39408, 39417, 39426, 39462, 39471, 39480, 39507, 39516, 39561, 39570, 39615, 39624, 39642, 39651, 39705, 39714, 39741, 39750, 39804, 39840

I've looked at the first number in this sequence so let's look at the last:$$39840 \rightarrow 3+9=8+4+0$$The sequence will eventually terminate because the largest number possible will contain all ten digits but what might this number be? For a start it can't contain all the digits from 0 to 9 because the sum of these digits is 45 and can't be divided into two equal parts. So we have to drop the 1 if looking for the largest possible number. I think the largest possible number is 985647320 where we have:$$985764320 \rightarrow 9+8+5=7+6+4+3+2+0$$It's also possible to swap the 5 on the left with the 32 on the right so that we get 983276540 so that we have:$$983276540 \rightarrow 9+8+3+2=7+6+5+4+0$$However, this number is smaller than the previous and so it is not the largest possible. What about five digit numbers that satisfy the following criteria:

• number has no repeating digits
• number can be split into two parts and a digit equation formed from the product of the digits on either side of the equal sign

In this case, between 27586 and 40000, there are only 38 numbers that qualify (permalink):

29136, 29163, 29316, 29361, 29613, 29631, 31426, 31462, 31629, 31692, 31846, 31864, 32649, 32694, 34126, 34162, 34216, 34261, 34612, 34621, 34689, 34698, 36129, 36192, 36219, 36249, 36291, 36294, 36489, 36498, 36912, 36921, 38146, 38164, 38416, 38461, 38614, 38641

Let's take the first number in this list, 29136, where we have:$$29136 \rightarrow 2 \times 9 = 1 \times 3 \times 6$$The last number in the list is 38641 where we have:$$38641 \rightarrow 3 \times 8 = 6 \times 4 \times 1$$Other variations on this theme are possible such as using the sum of squares of the digits. For example, let's propose the criteria:

• number has no repeating digits
• number can be split into two parts and a digit equation formed from the sum of the squares of the digits on either side of the equal sign

These criteria yield 48 numbers in the range from 27586 to 40000. These are (permalink):

27614, 27641, 27658, 27685, 27869, 27896, 28769, 28796, 29067, 29076, 29607, 29670, 29706, 29760, 30267, 30627, 31857, 31875, 32067, 32607, 32670, 34517, 34571, 35417, 35471, 36027, 36207, 36245, 36254, 36270, 36425, 36452, 36524, 36542, 38157, 38175, 39158, 39185, 39457, 39475, 39518, 39547, 39574, 39581, 39745, 39754, 39815, 39851

Let's take the first number in the list, 27614, as an example:$$27614 \rightarrow 2^2+7^2 = 6^2+1^2+4^2$$The last number in the list, 39851, can be split as follows:$$39851 \rightarrow 3^2+9^2=8^2+5^2+1^2$$More variations are possible of course but that will do for now. Forming digit equations from the digits of a number falls most definitely into the realm of recreational mathematics and is base-10 specific but it's an interesting mental exercise and what's wrong with having fun with numbers anyway.

These types of mental exercises, especially for children, can serve as an easy entry point to number theory after which they can embark on a deeper exploration of number properties that are not base-specific and that are intrinsic to the number itself, such as primeness and the sum of a number's divisors relative to the number itself that determines whether it is deficient, perfect or abundant.