Smith Number Subsets

Smith numbers have the property that the sums of their digits are equal to the sums of the digits of their prime factors with multiplicity. Based on that criterion, the number associated with my diurnal age today, 28174, is a Smith number:$$28174=2 \times 14087$$This number however, has a further property if you look at it closely. The number and its prime factors share the same digits with the exception of the zero. This qualifies it for membership in OEIS A176670:

A176670: composite numbers having the same digits as their prime factors (with multiplicity), excluding zero digits.

The initial members of the sequence are:

1111, 1255, 12955, 17482, 25105, 28174, 51295, 81229, 91365, 100255, 101299, 105295, 107329, 110191, 110317, 117067, 124483, 127417, 129595, 132565, 137281, 145273, 146137, 149782, 163797, 171735, 174082, 174298, 174793, 174982, 193117, 208174, 210181, 217894

The table below shows the details:

  number   digit sum   factors         sum of factors' digits

  1111     4           11 * 101        4
  1255     13          5 * 251         13
  12955    22          5 * 2591        22
  17482    22          2 * 8741        22
  25105    13          5 * 5021        13
  28174    22          2 * 14087       22
  51295    22          5 * 10259       22
  81229    22          29 * 2801       22
  91365    24          3 * 5 * 6091    24
  100255   13          5 * 20051       13
  101299   22          11 * 9209       22
  105295   22          5 * 21059       22
  107329   22          29 * 3701       22
  110191   13          101 * 1091      13
  110317   13          107 * 1031      13
  117067   22          167 * 701       22
  124483   22          281 * 443       22
  127417   22          47 * 2711       22
  129595   31          5 * 25919       31
  132565   22          5 * 26513       22
  137281   22          107 * 1283      22
  145273   22          53 * 2741       22
  146137   22          317 * 461       22
  149782   31          2 * 74891       31
  163797   33          3 * 71 * 769    33
  171735   24          3 * 5 * 107^2   24
  174082   22          2 * 87041       22
  174298   31          2 * 87149       31
  174793   31          47 * 3719       31
  174982   31          2 * 87491       31
  193117   22          113 * 1709      22
  208174   22          2 * 104087      22
  210181   13          101 * 2081      13
  217894   31          2 * 108947      31

What's interesting is that these same digits show up several times in the above table:

• 17482 which is a permutation of the digits of 28174
• 149782 which has the digit 9 added to the permuted digits
• 174082 which has the digit 0 added to the permuted digits
• 174298 which has the digit 9 added to the permuted digits
• 174982 which has the digit 9 added to the permuted digits
• 208174 which has the digit 0 inserted after the 2 in 28174

All these numbers are xenodromes meaning that they have no repeated digits.

The obverse of this is to find all Smith numbers that have NO digits in common with their prime factors. There are 72 of these in the range up 40000 (permalink):

4, 27, 58, 166, 454, 576, 588, 627, 648, 654, 666, 690, 706, 729, 1449, 1858, 1908, 2067, 2409, 2839, 4369, 4414, 4464, 4880, 4960, 5458, 5818, 5854, 6084, 6096, 6567, 6583, 6684, 6718, 6760, 6880, 7068, 7078, 7186, 8158, 8568, 8680, 8864, 8901, 9166, 9414, 9849, 10669, 10786, 10966, 14458, 14566, 14958, 15646, 15709, 15984, 16546, 16866, 17496, 17664, 17718, 17840, 18418, 18454, 19818, 20229, 20299, 22509, 26727, 33680, 33760, 33880

The table below shows the details:

  number   digit sum   factors              sum of factors' digits

  4        4           2^2                  4
  27       9           3^3                  9
  58       13          2 * 29               13
  166      13          2 * 83               13
  454      13          2 * 227              13
  576      18          2^6 * 3^2            18
  588      21          2^2 * 3 * 7^2        21
  627      15          3 * 11 * 19          15
  648      18          2^3 * 3^4            18
  654      15          2 * 3 * 109          15
  666      18          2 * 3^2 * 37         18
  690      15          2 * 3 * 5 * 23       15
  706      13          2 * 353              13
  729      18          3^6                  18
  1449     18          3^2 * 7 * 23         18
  1858     22          2 * 929              22
  1908     18          2^2 * 3^2 * 53       18
  2067     15          3 * 13 * 53          15
  2409     15          3 * 11 * 73          15
  2839     22          17 * 167             22
  4369     22          17 * 257             22
  4414     13          2 * 2207             13
  4464     18          2^4 * 3^2 * 31       18
  4880     20          2^4 * 5 * 61         20
  4960     19          2^5 * 5 * 31         19
  5458     22          2 * 2729             22
  5818     22          2 * 2909             22
  5854     22          2 * 2927             22
  6084     18          2^2 * 3^2 * 13^2     18
  6096     21          2^4 * 3 * 127        21
  6567     24          3 * 11 * 199         24
  6583     22          29 * 227             22
  6684     24          2^2 * 3 * 557        24
  6718     22          2 * 3359             22
  6760     19          2^3 * 5 * 13^2       19
  6880     22          2^5 * 5 * 43         22
  7068     21          2^2 * 3 * 19 * 31    21
  7078     22          2 * 3539             22
  7186     22          2 * 3593             22
  8158     22          2 * 4079             22
  8568     27          2^3 * 3^2 * 7 * 17   27
  8680     22          2^3 * 5 * 7 * 31     22
  8864     26          2^5 * 277            26
  8901     18          3^2 * 23 * 43        18
  9166     22          2 * 4583             22
  9414     18          2 * 3^2 * 523        18
  9849     30          3 * 7^2 * 67         30
  10669    22          47 * 227             22
  10786    22          2 * 5393             22
  10966    22          2 * 5483             22
  14458    22          2 * 7229             22
  14566    22          2 * 7283             22
  14958    27          2 * 3^3 * 277        27
  15646    22          2 * 7823             22
  15709    22          23 * 683             22
  15984    27          2^4 * 3^3 * 37       27
  16546    22          2 * 8273             22
  16866    27          2 * 3^2 * 937        27
  17496    27          2^3 * 3^7            27
  17664    24          2^8 * 3 * 23         24
  17718    24          2 * 3 * 2953         24
  17840    20          2^4 * 5 * 223        20
  18418    22          2 * 9209             22
  18454    22          2 * 9227             22
  19818    27          2 * 3^3 * 367        27
  20229    15          3 * 11 * 613         15
  20299    22          53 * 383             22
  22509    18          3^2 * 41 * 61        18
  26727    24          3 * 59 * 151         24
  33680    20          2^4 * 5 * 421        20
  33760    19          2^5 * 5 * 211        19
  33880    22          2^3 * 5 * 7 * 11^2   22

Four Special Xenodromes

The number associated with my diurnal age today (\( \textbf{28149} \)) has the property that it is a member of OEIS A365257. This sequence consists of numbers such that the five digits of the number and their four successive absolute first differences are all distinct. The digit 0 is excluded of course or else the condition cannot be met.$$ \underbrace{|2-8|}_{6} \, \underbrace{|8-1|}_{7} \, \underbrace{|1-4|}_{3} \, \underbrace{|4-9|}_{5}$$I've written about this sequence before in a post titled Very Special Five Digit Numbers. In the range up to 40000, the only 96 such numbers (with 0 excluded) are:

14928, 15829, 17958, 18259, 18694, 18695, 19372, 19375, 19627, 25917, 27391, 27398, 28149, 28749, 28947, 34928, 35917, 37289, 37916, 38926, 39157, 39578, 43829, 45829, 47289, 47916, 49318, 49681, 49687, 51869, 53719, 57391, 57398, 58926, 59318, 59681, 59687, 61973, 61974, 62983, 62985, 67958, 68149, 68749, 68947, 69157, 69578, 71952, 71953, 72691, 72698, 74619, 74982, 74986, 75193, 75196, 76859, 78259, 78694, 78695, 81394, 81395, 81539, 82941, 82943, 85179, 85629, 85971, 85976, 86749, 87269, 87593, 87596, 89372, 89375, 89627, 91647, 91735, 92658, 92834, 92851, 92854, 93518, 94182, 94186, 94768, 94782, 94786, 95281, 95287, 95867, 96278, 96815, 97158, 98273, 98274

This got me wondering as to how many numbers there were such that the digits of the numbers contain the non-prime digits (1, 4, 6, 8 and 9) and the absolute differences between successive pairs of digits are the prime digits 2, 3, 5 and 7. It turns out that there are only four such numbers (grouped into two palindromic pairs):$$ \begin{align} 18694 \rightarrow \underbrace{|1-8|}_{7} \, \underbrace{|8-6|}_{2} \, \underbrace{|6-9|}_{3} \, \underbrace{|9-4|}_{5}\\ \\49681 \rightarrow \underbrace{|4-9|}_{5} \, \underbrace{|9-6|}_{3} \, \underbrace{|6-8|}_{2} \, \underbrace{|8-1|}_{7}\\ \\94186 \rightarrow \underbrace{|9-4|}_{5} \, \underbrace{|4-1|}_{3} \, \underbrace{|1-8|}_{7} \, \underbrace{|8-6|}_{2} \\ \\68149 \rightarrow \underbrace{|6-8|}_{2} \, \underbrace{|8-1|}_{7} \, \underbrace{|1-4|}_{3} \, \underbrace{|4-9|}_{5} \end{align}$$

Testing For Simple Permutations

Having explained what a simple permutation is in my previous post, I thought that testing the number associated with my diurnal age would be a good idea. Here is the output for today's number \( \textbf{28146} \) (permalink):

Simple Permutation Analyzer initialized...

--- Analyzing Number: 28146 ---

Intermediate Stage:

  Digits ranked descending: [8, 6, 4, 2, 1]

  Rank mapping (1=highest): {8: 1, 6: 2, 4: 3, 2: 4, 1: 5}

  Normalized permutation:   [4, 1, 5, 3, 2]

Result: False

Reason: Failed: Sub-block [3, 2] from digits [4, 6] forms an interval.

Alternative: Try 28416 instead. It is a simple permutation using your digits.

Having found that the digits 28146 do not form a simple permutation, the program finds a permutation that is simple, namely 28416. I tried entering a number with repeated digits and the number was regarded as invalid. I asked Gemini why:

The Definition of a Strict Permutation

The primary reason duplicate digits are rejected is rooted in the mathematical definition of a permutation. In combinatorics, a standard permutation is strictly defined as a rearrangement of a set of distinct, distinguishable items. Every item in the set must be unique.

When you introduce duplicate items (like the number 88234), you are no longer working with a standard set. You are working with what mathematicians call a multiset. While you can arrange the elements of a multiset, the rules, formulas, and structural properties change entirely.

The Problem of Ambiguity

Permutation theory relies on knowing exactly how many unique arrangements can be made. If all elements are distinct, a sequence of length n will always have n! (n factorial) possible arrangements.

However, duplicates introduce ambiguity. Imagine the number 122. Let's label the identical twos to tell them apart: 1, 2_a, 2_b. If we swap the two 2s, we get a structurally identical number:

• Arrangement A: 1, 2_a, 2_b (Displays as 122)
• Arrangement B: 1, 2_b, 2_a (Displays as 122)

Because swapping identical digits does not create a new, distinct sequence, the total number of unique arrangements drops. The mathematical framework built to analyze the structure of permutations assumes that every single shift in position creates a fundamentally new sequence. Duplicates break this assumption.

How Duplicates Break the "Simple" Math

Beyond the theoretical definition, duplicate digits physically break the mathematical logic used to define a simple permutation.

Recall the formula we used in the algorithm to test for intervals: max(block) - min(block) == length - 1. This formula only works because it assumes that an interval of distinct integers will have a predictable span. For example, a block of three distinct, consecutive numbers like {4, 5, 6} will always have a span of 2 (because 6 - 4 = 2, and the length 3 minus 1 is 2).

If we allow duplicates, this logic collapses. Consider a block of two identical numbers: {5, 5}.

• The length of the block is 2.
• The maximum value is 5.
• The minimum value is 5.
• max - min is 0.

The formula expects a span of 1 for a block of length 2. Because the span is 0, the math fails to recognize what is happening. Furthermore, the entire concept of a "continuous range of numbers" becomes nonsensical when numbers repeat. Does {5, 5} constitute an interval? What about {4, 5, 5, 6}?

Because the definition of simple permutations was explicitly designed to map the structural scattering of distinct integers, attempting to apply it to repeating digits forces the math to evaluate scenarios it was never designed to handle.

Fair enough. I've incorporated this algorithm into my daily number analysis. The no repeated digits restriction means that only xenodromes are acceptable input.

Xenodrome Pairs, Triplets Etc.

\( \textbf{27815} \), the number associated with my diurnal age today, is one of those numbers for which an interesting property is hard to come by. However, I did notice that the number was a xenodrome in both base 10 and base 9 (42135). A little further investigation revealed that 27816 also shared this property. That got me thinking about how often pairs of such numbers occur. Now I've written about xendromes before in posts titled Xenodromes and Xenodrome Probabilities in which I've looked at numbers that remain xenodromes in various bases but so far I've not looked at groupings of numbers by pairs, triplets, quadruplets etc. with properties relating to xenodromes.

In my Bespoken for Sequences database, I've listed the following sequence of numbers:$$ \begin{align} \textbf{Smaller of a pair of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$Between 27815 and 40000, there are 282 numbers with this property (permalink):

27815, 27834, 27835, 27845, 27860, 27950, 27953, 27960, 28013, 28016, 28134, 28169, 28195, 28196, 28314, 28346, 28356, 28364, 28395, 28456, 28495, 28509, 28536, 28563, 28573, 28590, 28609, 28670, 28914, 28934, 28935, 28963, 29015, 29016, 29053, 29075, 29084, 29085, 29103, 29134, 29147, 30147, 30148, 30156, 30157, 30186, 30194, 30195, 30196, 30457, 30458, 30467, 30528, 30548, 30561, 30591, 30691, 30724, 30725, 30751, 30764, 30814, 30825, 30851, 30924, 30925, 31024, 31025, 31048, 31057, 31086, 31094, 31095, 31096, 31097, 31257, 31258, 31259, 31267, 31268, 31284, 31294, 31475, 31586, 31604, 31607, 31608, 31806, 31824, 31825, 31826, 31859, 31905, 31906, 31907, 32018, 32104, 32108, 32159, 32189, 32496, 32508, 32509, 32546, 32580, 32589, 32590, 32607, 32608, 32609, 32648, 32657, 32670, 32689, 32690, 32697, 32907, 32947, 32960, 32964, 34027, 34058, 34085, 34086, 34091, 34095, 34127, 34157, 34158, 34175, 34185, 34206, 34215, 34275, 34278, 34279, 34296, 34297, 34560, 34567, 34568, 34569, 34578, 34620, 34650, 34658, 34761, 34785, 34815, 34820, 34905, 34917, 34926, 35016, 35017, 35047, 35097, 35169, 35196, 35197, 35208, 35216, 35217, 35479, 35486, 35496, 35641, 35809, 35826, 35890, 35891, 35916, 35917, 35946, 35961, 35970, 35971, 35980, 35981, 36018, 36027, 36028, 36208, 36209, 36214, 36218, 36270, 36278, 36279, 36280, 36289, 36290, 37195, 37204, 37245, 37285, 37294, 37295, 37458, 37459, 37485, 37495, 37519, 37520, 37528, 37568, 37819, 37820, 37845, 37864, 37890, 37891, 37920, 37945, 37964, 37981, 38015, 38024, 38045, 38046, 38064, 38105, 38106, 38124, 38145, 38159, 38169, 38205, 38206, 38240, 38249, 38250, 38259, 38260, 38405, 38406, 38415, 38420, 38469, 38649, 38670, 38674, 38694, 38701, 38720, 38724, 38751, 38760, 38904, 38916, 39015, 39016, 39017, 39024, 39025, 39026, 39124, 39125, 39126, 39186, 39205, 39206, 39207, 39215, 39216, 39240, 39260, 39481, 39485, 39540, 39541, 39571, 39580, 39604, 39701, 39715, 39724, 39764, 39805, 39814, 39845, 39846

There are 69  triplets in the same range (permalink):$$ \begin{align} \textbf{Smallest of a triplet of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$27834, 28195, 28934, 29015, 29084, 30147, 30156, 30194, 30195, 30457, 30724, 30924, 31024, 31094, 31095, 31096, 31257, 31258, 31267, 31607, 31824, 31825, 31905, 31906, 32508, 32589, 32607, 32608, 32689, 34085, 34157, 34278, 34296, 34567, 34568, 35016, 35196, 35216, 35890, 35916, 35970, 35980, 36027, 36208, 36278, 36279, 36289, 37294, 37458, 37519, 37819, 37890, 38045, 38105, 38205, 38249, 38259, 38405, 39015, 39016, 39024, 39025, 39124, 39125, 39205, 39206, 39215, 39540, 39845

An example is the xenodrome 27834 with 27835 and 27836 also xenodromes. The base 9 equivalents 42156, 42157 and 42158 are also xenodromes.

There are 13 quadruplets in the same range (permalink):$$ \begin{align} \textbf{Smallest of a quaduplet of consecutive numbers} \\ \textbf{that are xenodromes in base 10 and base 9} \end{align} $$30194, 31094, 31095, 31257, 31824, 31905, 32607, 34567, 36278, 39015, 39024, 39124, 39205

An example is the xenodrome 30194 with 30195, 30196 and 30197 also xenodromes. The base 9 equivalents 45368, 45370, 45371 and 45372 are also xenodromes. 

There is only one \( \textbf{quintuplet}\) in the range and that is 31094. Here we see that 31094, 31095, 31096, 31097 and 31098 are all xenodromes as are their base 9 equivalents 46578, 46580, 46581, 46582 and 46583.

Blast From The Past

Looking back over my old tweets on Twitter (as it was once called), I noticed that my first reference to a day count occurred on August 11th 2013. See Figure 1.

Figure 1

It's a fairly unimpressive tweet that, from a mathematical perspective, simply notes that 23506 has four distinct prime factors. I wasn't aware of the Online Encyclopedia of Integer Sequences or OEIS back then, not Numbers Aplenty or most other resources that I now use. The OEIS informs us that 23506 is a member of A029793:

 A029793    Numbers \(k\) such that \(k\) and \(k^2\) have the same set of digits.

This is because \(23506^2=552532036\) and there are not many numbers in the range up to 40000 that have this property. The numbers that do are: 

0, 1, 10, 100, 1000, 4762, 4832, 10000, 10376, 10493, 11205, 12385, 12650, 14829, 22450, 23506, 24605, 26394, 34196, 36215

From Numbers Aplenty we learn that 23506 has properties that make it

• a self number because there is no number that added to its sum of digits gives 23506
  
• an untouchable number because it is not equal to the sum of proper divisors of any number

My own algorithm tells me that 23506 is:

• a xenodrome in base 9: 35217 and in base 10: 23506 because all digits are different
  
• an attractor because its sum of even and odd digits are the same (8)

These are just a few of the special properties that 23506 possesses. I've come a long way since those early days.

An Interesting Triple 7 Number

Today I turned \( \textbf{27717} \) days old and this number has a plethora of interesting properties that deserve a special mention and thus a dedicated post. Here are some of those properties.

• \( \textbf{27717} \) is a so-called Lucky Cube, meaning it is a number whose cubes contain the digit sequence “888”, here:$$27717^3 = 21293088810813$$The numbers that satisfy from 27717 to 40000 are: 27717, 27942, 27973, 28192, 28442, 28484, 28692, 28740, 28942, 29079, 29192, 29354, 29387, 29391, 29418, 29420, 29442, 29491, 29642, 29692, 29942, 29989.
  
  
• \( \textbf{27717} \) is the lesser of a pair of adjacent composite numbers such that both are only one step away from their home primes. Here: 
• \(27717 = 3 \times 9239 \rightarrow 39239\)
• \(27718 = 2 \times 13859 \rightarrow 213859\)
  
  
• \( \textbf{27717} \) is a number such that n + POD(n) and n - POD(n) are both prime (where POD stands for Product Of Digits). Here we have POD = 686:
• \(27717 + 686 = 28403\) which is a prime number
• \(27717 - 686 = 27031\) which is a prime number
  
  
• \( \textbf{27717} \) is an interprime number because it is at equal distance from the previous prime (27701) and the next prime (27733).
  
  
• \( \textbf{27717} \) is a number whose sum of divisors has prime factors (ignoring multiplicity) that multiply to the factorial 2310 where
  
  \(2310= 2 \times 3 \times 5 \times 7 \times 11\)
  
  Here 27717 has a sum of divisors 36960 and
  
  \(36960= 2^5 \times 3 \times 5 \times 7 \times 11\)
  
  but also forms a consecutive pair with 27718 because its sum of the divisors is 41580 and
  
  \(41580= 2^2 \times 3^3 \times 5  \times 7 \times11\)
  
  See blog post Primorials and the Sigma Function.
  
  
• \( \textbf{27717} \) is the TENTH member of an interesting number chain (which is base independent):
  
• \(27708 = 12 \times 2309\)
• \(27709 = 11 \times 2519\)
• \(27710 = 10 \times 2771\)
• \(27711 = 9 \times 3079\)
• \(27712 = 8 \times 3464\)
• \(27713 = 7 \times 3959\)
• \(27714 = 6 \times 4619\)
• \(27715 = 5 \times 5543\)
• \(27716 = 4 \times 6929\)
• \(27717 = 3 \times 9239\)
• \(27718 = 2 \times 13859\)
  

See blog post Count Down Number Chains  

 

• \( \textbf{27717} \) is a cyclic number.
  
  
• \( \textbf{27717} \) is a xenodrome in base 9 : 42016. See blog post Xenodromes.
  
  
• \( \textbf{27717} \) is a number that does not reach a palindrome after 2001 cycles of the reverse and add algorithm.
  
  
• \( \textbf{27717} \) is a D-number meaning it is a number \(n > 3\) such that n divides \( k^{n-2}- k\) for all \(1 < k < n\) relatively prime to \(n\).
  
  
• \( \textbf{27717} \) can be rendered as a digit equation as follows: \(2 - \dfrac{7}{7} = 1 ^ 7\)

Xenodrome Probabilites

I posted about xenodromes in an eponymous post on the 20th October 2024. In this current post, I want to look at the probabilities of xenodromes occurring in the various number bases. Let's start with base 16 where we have 16 digits to choose from:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, a, b, c, d, e, f

An example of a five digit xenodrome in base 16 would be 56a3f where:$$56a3f_{16}=354879_{10}$$What I found was the following (permalink):

There are 491400 permutations of 5 digits in base 16 so that all are distinct with no leading zero. There are a total of 983040 possible permutations of 5 digits in base 16 with no leading zero. Probability of a 5 digit xenodrome in base 16 is thus 0.4999 or 50%.

The probability is thus effectively 50%. However, the five digit decimal numbers that I deal with when investigating the numbers associated with my diurnal age (10000 to 40000) require only four hexadecimal digits for their representation and so:

There are 40950 permutations of 4 digits in base 16 so that all are distinct with no leading zero. There are a total of 61440 possible permutations of 4 digits in base 16 with no leading zero. Probability of a 4 digit xenodrome in base 16 is thus 0.6665 or 67%.

So 5 digit base 10 numbers in the range between 10000 and 40000 will have a 2/3 chance of being xenodromic in base 16. 

Let's look at the other bases now. Here's the results for base 15 and 5 digits:

There are 336336 permutations of 5 digits in base 15 so that all are distinct with no leading zero. There are a total of 708750 possible permutations of 5 digits in base 15 with no leading zero. Probability of a 5 digit xenodrome in base 15 is thus 0.4745 or 47%.

 For base 14 and 5 digits:

There are 223080 permutations of 5 digits in base 14 so that all are distinct with no leading zero. There are a total of 499408 possible permutations of 5 digits in base 14 with no leading zero. Probability of a 5 digit xenodrome in base 14 is thus 0.4467 or 45%.

For base 13 and 5 digits:

There are 142560 permutations of 5 digits in base 13 so that all are distinct with no leading zero. There are a total of 342732 possible permutations of 5 digits in base 13 with no leading zero. Probability of a 5 digit xenodrome in base 13 is thus 0.4160 or 42%.

For base 12 and 5 digits: 

There are 87120 permutations of 5 digits in base 12 so that all are distinct with no leading zero There are a total of 228096 possible permutations of 5 digits in base 12 with no leading zero Probability of a 5 digit xenodrome in base 12 is thus 0.3819 or 38%.

For base 11 and 5 digits: 

There are 50400 permutations of 5 digits in base 11 so that all are distinct with no leading zero There are a total of 146410 possible permutations of 5 digits in base 11 with no leading zero Probability of a 5 digit xenodrome in base 11 is thus 0.3442 or 34%.

For base 10 and 5 digits:

There are 27216 permutations of 5 digits in base 10 so that all are distinct with no leading zero There are a total of 90000 possible permutations of 5 digits in base 10 with no leading zero Probability of a 5 digit xenodrome in base 10 is thus 0.3024 or 30%.

So the probability of a five digit xenodrome in base 10 is 30%. We may as well continue for the lower bases.

For base 9 and 5 digits:

There are 13440 permutations of 5 digits in base 9 so that all are distinct with no leading zero. There are a total of 52488 possible permutations of 5 digits in base 9 with no leading zero. Probability of a 5 digit xenodrome in base 9 is thus 0.2561 or 26%.

For base 8 and 5 digits: 

 There are 5880 permutations of 5 digits in base 8 so that all are distinct with no leading zero. There are a total of 28672 possible permutations of 5 digits in base 8 with no leading zero. Probability of a 5 digit xenodrome in base 8 is thus 0.2051 or 21%.

For base 7 and 5 digits:

There are 2160 permutations of 5 digits in base 7 so that all are distinct with no leading zero. There are a total of 14406 possible permutations of 5 digits in base 7 with no leading zero. Probability of a 5 digit xenodrome in base 7 is thus 0.1499 or 15%. 

For base 6 and 5 digits:

There are 600 permutations of 5 digits in base 6 so that all are distinct with no leading zero. There are a total of 6480 possible permutations of 5 digits in base 6 with no leading zero. Probability of a 5 digit xenodrome in base 6 is thus 0.09259 or 9%. 

For base 5 and 5 digits:

There are 96 permutations of 5 digits in base 5 so that all are distinct with no leading zero. There are a total of 2500 possible permutations of 5 digits in base 5 with no leading zero. Probability of a 5 digit xenodrome in base 5 is thus 0.03840 or 4%.

Of course, there are no 5 digit xenodromes in base 4 so we need to switch to four digit numbers. So for base 4 and 4 digits:

There are 18 permutations of 4 digits in base 4 so that all are distinct with no leading zero. There are a total of 192 possible permutations of 4 digits in base 4 with no leading zero. Probability of a 4 digit xenodrome in base 4 is thus 0.09375 or 9%.

Similarly there no 4 digit xenodromes in base 3 so we need to switch to three digits. For base 3 and 3 digits:

There are 4 permutations of 3 digits in base 3 so that all are distinct with no leading zero. There are a total of 18 possible permutations of 3 digits in base 3 with no leading zero. Probability of a 3 digit xenodrome in base 3 is thus 0.2222 or 22%.

Finally, for base 2 and 2 digits we have:

There is 1 permutation of 2 digits in base 2 so that all are distinct with no leading zero. There are a total of 2 possible permutations of 2 digits in base 2 with no leading zero. Probability of a 2 digit xenodrome in base 2 is thus 0.5000 or 50%.

Here is the permalink again to carry out these calculations. Just adjust for the number of digits and the base. 

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