888 Revisited

In November of 2021, I posted about 888 from a mathematical and non-mathematical perspective. Mathematically, I noted that:

• 888 is the smallest cube in which each digit occurs exactly three times. The list up to one million of such numbers is (permalink):

888, 56592, 58524, 65577, 70869, 78183, 496941, 512427, 516267, 517461, 557949, 565920, 581421, 585558, 661959, 711828, 713772, 723627, 724983, 733053, 739563, 764472, 781830, 877242, 988458

• 888 is the only cube in which three digits occur three times. For example, the next number in the previous sequence (56592) has a cube of 181244621426688 but there are four digits that occur three times.
  
  
• 888 the smallest multiple of 24 whose digit sum is 24 and, as well as being divisible by its digit sum, it is divisible by all of its digits.
  
  
• 888 and 24 show up again in the former's membership of OEIS 236661 where 888 counts the number of partitions of 24 that have a standard deviation greater than 2. Permalink.
  
  
• Other properties of 888 include its being a happy, Harshad, Moran, nude, strobogrammatic, modest, congruent, amenable, practical, abundant, pseudoperfect and Zumkeller number (see Numbers Aplenty).
  
  
• The 8's are involved in 888 again thanks to its membership of OEIS A127335.
  
  

A127335

Numbers that are the sum of 8 successive primes.

 The sequence runs:

77, 98, 124, 150, 180, 210, 240, 270, 304, 340, 372, 408, 442, 474, 510, 546, 582, 620, 660, 696, 732, 768, 802, 846, 888

The eight successive primes in the case of 888 are 97, 101, 103, 107, 109, 113, 127 and 131 with an average of 111. Both 111 and 888 are of course repdigits along with the infamous 666 or number of the beast.

• 888 arises in the context of aliquot sequences via OEIS A014360:
  
  

A014360

Aliquot sequence starting at 552.

The sequence begins:

552, 888, 1392, 2328, 3552, 6024, 9096, 13704, 20616, 30984, ...

To quote from Wolfram Alpha:

It has not been proven that all aliquot sequences eventually terminate and become periodic. The smallest number whose fate is not known is 276. There are five such sequences less than 1000, namely 276, 552, 564, 660, and 966, sometimes called the "Lehmer five". 

In this post, I'll revisit 888 but I'll be doing so via 28192, the number associated with my diurnal age today. This number has the interesting property that its square and cube both contain the digit sequence 888:$$ \begin{align} 28192^2 &= 7947\textbf{888}64 \\ 28192^3 &= 22406687653\textbf{888} \end{align} $$There are only four such numbers in the range up to 40000 and they are 3192, 28192, 31920 and 33878. One of these, 31920, is simply a derivative of 3192. Of these four numbers, it is only 28192 that contains three eights when expressed in terms of its prime factors:$$2\textbf{8}192 = 2^5 \times \textbf{88}1$$It could even be written as \(2^2 \times \textbf{8} \times \textbf{88}1 \) so it is very giving in terms of its eightness!

What about other numbers in the range up to 40000 that display three digit repdigits when squared and cubed? Here are the numbers that satisfy:

For repdigit \( \textbf{111}\), there is only one number that satisfies:

• \(10558^2= \textbf{111}471364 \text{ and } 10558^3 =117691466\textbf{111}2\)

For repdigits \( \textbf{222} \) and \( \textbf{333} \), no numbers satisfy but for repdigit \( \textbf{444}\) we have:

• \(6962^2 = 48469\textbf{444} \text{ and } 6962^3 = 337\textbf{444}269128\)
• \(12038^2 = 144913\textbf{444} \text{ and } 12038^3 = 17\textbf{444}68038872\)
• \(21081^2 = \textbf{4444}08561 \text{ and } 21081^3 = 936857687\textbf{444}1\)
• \(32538^2 = 1058721\textbf{444} \text{ and } 32538^3 = 3\textbf{444}8678344872\)
• \(37808^2 = 1429\textbf{444}864 \text{ and } 37808^3 = 540\textbf{444}51418112\)

For repdigit \( \textbf{555} \), again only one number satisfies:

• \(38152^2 =14\textbf{555}75104 \text{ and } 38152^3 = \textbf{555}33101367808\)

For the famous repdigit \( \textbf{666} \), we have two numbers that satisfy with some extra 6's thrown in for the case of \(30605\):

• \(17767^2 = 315\textbf{666}289 \text{ and } 17767^3 = 560844295\textbf{666}3\)
• \(30605^2 = 93\textbf{6666}025 \text{ and }30605^3 = 28\textbf{66666}3695125\)

For the repdigit \( \textbf{777} \), again two numbers satisfy:

• \( 18924^2 = 35811\textbf{777}6 \text{ and }18924^3= 6\textbf{777}020793024 \)
• \( 34753^2 =120\textbf{777}1009 \text{ and } 34753^3 = 41973665875\textbf{777} \)

For the repdigit \( \textbf{999} \) we have a bonanza so I'll just list the numbers and show one example:

\( 9997, 9998, 9999, 19998, 19999, 29999, 38729, 39999 \)

• \(9997^2 = \textbf{999}40009 \text{ and } 9997^3 = \textbf{999}100269973\)

Repdigits

On May 4th 2022, I turned 26694 days old and one of the properties of 26694 is that is a member of OEIS A167782: numbers that are repdigits with length > 2 in some base. Hmmm, but what base? Looking at the table from Numbers Aplenty, it can be seen that none of the bases from 2 to 16 satisfy (see Table 1).

Table 1

Testing out the bases from 17 to 36 (permalink), it can be seen that again there are no repdigits (see Table 2).

Table 2

The reason for stopping at base 36 is that we have run out of letters of the alphabet and need to resort to an alternative system for representing numbers in higher bases. Now I resorted to trial and error. Base 37 didn’t satisfy but base 38 did. I found that \( 18 \times 38^2+18 \times 38 + 18 = 26694 \) which we can write as \(18.18.18_{36} \). Of course, I got lucky. The number may have been a repunit in a much higher base. However, it’s easy to write a program to handle these higher bases and avoid wasting time on manual calculation (permalink). See Table 3.

Table 3

The table above shows representations for bases up to 60 and if nothing showed up in this range then calculations could be extended until the first number in the triplet reaches zero, in which case the length is equal to 2 and the number does not meet the criterion. For 26694, this occurs at base 164 where we find that \(0 \times 164^2+ 162 \times 164 + 126 = 26694\) which we can represent as \( 162.126_{164} \).

The initial members of OEIS A167782 are: 0, 7, 13, 15, 21, 26, 31, 40, 42, 43, 57, 62, 63, 73, 80, 85, 86, 91, 93, 111, 114, 121, 124, 127, 129, 133, 146, 156, 157, 170, 171, 172, 182, 183, 211, 215, 219, 222, 228, 241, 242, 255, 259, 266, 273, 285, 292, 307, 312, 314, 333, 341, 342, 343, 364, 365, 366. The accompanying comments can be found in the OEIS entry: definition requires “length > 2” because all numbers n > 2 are trivially represented as “11” in base n-1. 0 included at the suggestion of Franklin T. Adams-Watters (and others) as 0 = 000 in any base.

The 10,000th such number in the sequence is 583,744 which means that the percentage of such numbers up to that limit is about 1.713% or a little over 17 per thousand. Let’s not confuse repdigits with repunits:

In recreational mathematics, a repunit is a number like 11, 111, or 1111 that contains only the digit 1 — a more specific type of repdigit. The term stands for repeated unit and was coined in 1966 by Albert H. Beiler in his book Recreations in the Theory of Numbers. Wikipedia

All repdigits are multiples of repunits e.g. 666 is a multiple of the repunit 111. 

St. Ives

 

As I was going to St. Ives,

I met a man with seven wives,

Each wife had seven sacks,

Each sack had seven cats,

Each cat had seven kits:

Kits, cats, sacks, and wives,

How many were there going to St. Ives? 

The traditional understanding of this rhyme is that only one is going to St. Ives—the narrator. All of the others are coming from St. Ives. The trick is that the listener assumes that all of the others must be totalled up, forgetting that only the narrator is said to be going to St. Ives. If everyone mentioned in the riddle were bound for St. Ives, then the number would be 2,802: the narrator, the man and his seven wives, 49 sacks, 343 cats, and 2,401 kits. Wikipedia.

The progression 7, 49, 343 and 2401 corresponds to successive powers of seven: \(7^1, 7^2, 7^3\) and \(7^4\). The St. Ives rhyme came to mind because today I turned 26411 days old and this number has the interesting property that it can be written:$$26411=7 \times 7 \times 7 \times 77$$This property qualifies it for membership in OEIS A161145:

A161145

Numbers which can be expressed as the product of numbers made of only sevens.

The members of the sequence, up to 100,000 are:

1, 7, 49, 77, 343, 539, 777, 2401, 3773, 5439, 5929, 7777, 16807, 26411, 38073, 41503, 54439, 59829, 77777

Of course, 7 wives, 49 sacks, 343 cats and 2401 kittens make an appearance in this sequence along with 26411. Clearly, I won't be celebrating number 38073 as this corresponds to Sunday, June 29th 2053 when I would be 104 years old. 

I devised a general purpose algorithm in SageMath that will generate not only the seven sequence but sequences for all digits between 2 and 9 inclusive. Here is its permalink. Applying this algorithm, I was able to generate the beginning members of OEIS A161140:

A161140

Numbers which can be expressed as the product of numbers made of only twos.

Unfortunately the calculation timed out for the range up to 100,000 so I needed to restrict the range to 30,000. The members up to that point are:

1, 2, 4, 8, 16, 22, 32, 44, 64, 88, 128, 176, 222, 256, 352, 444, 484, 512, 704, 888, 968, 1024, 1408, 1776, 1936, 2048, 2222, 2816, 3552, 3872, 4096, 4444, 4884, 5632, 7104, 7744, 8192, 8888, 9768, 10648, 11264, 14208, 15488, 16384, 17776, 19536, 21296, 22222, 22528, 28416

Let's take 28416 as an example:$$28416=2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 2 \times 222$$Applying the algorithm so that all numbers are made up to threes generates OEIS A161141:

A161141

Numbers which can be expressed as the product of numbers made of only threes.

The members, up to 100,000, are:

1, 3, 9, 27, 33, 81, 99, 243, 297, 333, 729, 891, 999, 1089, 2187, 2673, 2997, 3267, 3333, 6561, 8019, 8991, 9801, 9999, 10989, 19683, 24057, 26973, 29403, 29997, 32967, 33333, 35937, 59049, 72171, 80919, 88209, 89991, 98901, 99999

Let's take 35937 as an example:$$35937=33 \times 33 \times 33$$Moving on to all fours (so to speak), we get OEIS A161142:

A161142

Numbers which can be expressed as the product of numbers made of only fours.

 The members, up to 100,000, are:

1, 4, 16, 44, 64, 176, 256, 444, 704, 1024, 1776, 1936, 2816, 4096, 4444, 7104, 7744, 11264, 16384, 17776, 19536, 28416, 30976, 44444, 45056, 65536, 71104, 78144, 85184

Let's take 17776 as an example:$$17776=4 \times 4444$$Moving on again to all fives, we get OEIS A161143:

A161143

Numbers which can be expressed as the product of numbers made of only fives.

The members, up to 100,000, are:

1, 5, 25, 55, 125, 275, 555, 625, 1375, 2775, 3025, 3125, 5555, 6875, 13875, 15125, 15625, 27775, 30525, 34375, 55555, 69375, 75625, 78125

Let's take 34375 as an example:$$34375=5 \times 5 \times 5 \times 5 \times 55$$Moving on further to all sixes, we get OEIS A161144:

A161144

Numbers which can be expressed as the product of numbers made of only sixes.

 The members up to 100,000, are:

1, 6, 36, 66, 216, 396, 666, 1296, 2376, 3996, 4356, 6666, 7776, 14256, 23976, 26136, 39996, 43956, 46656, 66666, 85536

Let's take 43956 as an example:$$43956=66 \times 666$$Moving on to all eights, we get OEIS A161146:

A161146

Numbers which can be expressed as the product of numbers made of only eights.

 The members, up to 100,000, are:

1, 8, 64, 88, 512, 704, 888, 4096, 5632, 7104, 7744, 8888, 32768, 45056, 56832, 61952, 71104, 78144, 88888

Let's use 71104 as an example:$$71104=8 \times 8888$$Moving on to all nines, we get OEIS A161147:

A161147

Numbers which can be expressed as the product of numbers made of only nines.

The members, up to 100,000, are:

1, 9, 81, 99, 729, 891, 999, 6561, 8019, 8991, 9801, 9999, 59049, 72171, 80919, 88209, 89991, 98901, 99999

Let's use 59049 as an example:$$59049=9 \times  9  \times 9 \times 9  \times 9$$

Osculators

In this post I'm describing a very interesting method of determining the divisors of any given number that I came across in a post by Sohel Sahoo:

All credit is given to Sohel Sahoo for his method and all I've done in this post is to rephrase his method so that it is easier for me to understand. 

The assertion is made that there exist two specific numbers (called osculators: one positive, the other negative) for any divisor. The divisor's positive or the negative osculator can be used to determine if a given number has that particular divisor. The sum of the absolute values of the osculators is equal to the divisor.

Procedures to find the osculators of a given divisor

(7 and 13 will be used as examples):

1. If the divisor is a single digit, work with the smallest multiple that has two digits.
   
   In the case of 7, the smallest two digit multiple is 14 so we work with that.
   
   In the case of 13, there is no need to modify it.
   
   
2. Let the osculator be \(x\) and multiply the unit digit of the divisor by \(x\).
   
   In the case of 14, this gives \(4x\).
   
   In the case of 13 this gives \(3x\).
   
   
3. Add the result obtained in step 2 to the remaining digits to obtain an expression in \(x\).
   
   In the case of 14 this gives the expression \(1+4x\).
   
   In the case of 13 this gives the expression \(1+3x\).
   
   
4. Find the smallest positive and negative values of \(x\) that make the expression divisible by the divisor. These are the positive and negative osculators for the divisor.
   
   In the case of 7, \(1+4 \times 5=21\) and \(1+4 \times -2 =-7\) and so the osculators are 5 and -2. Note that |5|+|-2|=7.
   
   In the case of 13, \(1 + 3 \times 4=13\) and \(1+ 3 \times -9 = -26\) and so the osculators are 4 and -9. Note that |4|+|-9|=13.

Doing this for other divisors generates the table shown below:

Number	Positive Osculator	Negative  Osculator
3	1	-2
7	5	-2
9	1	-8
11	10	-1
13	4	-9
17	12	-5
19	2	-17
21	19	-2
23	7	-16
27	19	-8
29	3	-26
31	28	-3
33	10	-23
37	26	-11
39	4	-35
41	37	-4
43	13	-30
47	33	-14
49	5	-44
51	46	-5
53	16	-37
57	40	-17
59	6	-53
61	55	-6
63	19	-44
67	47	-20
69	7	-62
71	64	-7
73	22	-51
77	54	-23
79	8	-71
81	73	-8
83	25	-58
87	61	-26
89	9	-80

Some tips for remembering the osculators:

Any divisor ending in 9 can generally be written as \(a9\).

Let the osculator be \(x\).

According to rules the expression will be \(9x+a\).

In decimal representation we have:

\(a9=10a+9=9a+a+9=9a+9+a=9(a+1)+a\)

Obviously, the positive osculator is (\(a+1\)). Henceforth, owing to this, the positive osculator for divisor numbers ending in 9 is just one more than its previous digits.

1. For 9, 19, 29, 39 etc.(all ending in 9), the positive osculators are 1,2,3,4 etc.
   
   
2. For 3, 13, 23, 33 etc. (all ending in 3) multiply them by 3 in order to get 9 in the unit place as 9, 39, 69, 99 etc. Thus you get 1, 4, 7, 10 etc. as positive osculators.
   
   
3. For 7, 17, 27, 37 etc.(all ending in 7) multiply them by 7 so as to attain 9 in unit place like 49, 119, 189, 259 etc. Hence you get 5, 12, 19, 26 etc. as positive osculators.
   
   
4. For 1, 11, 21, 31 etc. (all ending in 1) multiply them by 9 thereof attain 9 as last digit as 9, 99, 189, 279 etc. So, you get positive osculators viz. 1, 10, 19, 28 etc.

Special Divisibility Rule:

If any number is made by repeating a digit 6 times then the no. will be divisible by 3, 7, 11, 13, 21, 37, 77, 91, 143 and 1001 e.g. 111111, 222222 and 333333 are divisible by these numbers.

Formula for using the osculators to test divisibility:

We'll use 69125 and divisibility by 7 as an example to illustrate the use of the formula. We'll use both the positive and negative osculators (although in practice, only one is needed):

1. Firstly, multiply the osculator of the divisor by the unit digit of the number that is being tested for divisibility by that divisor.
   
   In the case of 69125 being test for divisibility by 7, this means 5 x 5 = 35 using the positive osculator of 5 and 5 x -2 = 10 using the negative osculator of -2.
   
   
2. Add the result so obtained on multiplication to the remaining of digits if using the positive osculator, subtract the result if using the negative osculator.
   
   Thus 69125 --> 6912 + 25 = 6937 or 6912 - 10 = 6902.
   
   
3. Repeat processes 1 and 2 until the number is small enough to be recognised as a multiple of the divisor or not. 
   
   Thus 6937 --> 693 + 35 = 728 --> 72 + 40 = 112 --> 11 + 10 --> 21 which is 7 x 3.
   6902 --> 690 - 4 = 686 --> 68 -12 = 56 which is 7 x 8.
   
   
4. If the result is a multiple of the divisor then the divisor is confirmed.
   
   69125 does reduce to a multiple of 7 and so 7 is a divisor. It can also be seen that using the smaller of the two osculators (ignoring signs) makes for easier calculations.

Another example:

What happens if 69125 were tested for divisibility by 13? We'll use the 4 as the osculator as it is the smaller of the two.

69125 --> 6912 - 20 = 6892 --> 689 - 8 = 681 --> 68 - 4 = 64 which is clearly not a multiple of 13 and so we conclude that 69125 is not divisible by 13.

Usefulness of Divisibility Tests

One can argue that learning divisibility tests like this is pointless in this technological age but I think such mental activities combat mental decline (I am a septuagenarian so that's important) and reduce our reliance on technology by making us more confident and self-sufficient (as a counterbalance to the encroachments of AI).

Note on Osculators and Osculation

It should be noted that the terms osculator and osculation have special meanings in Vedic Mathematics. An osculator is an algorithm for performing osculation while osculation means the determination of whether a number is divisible by another by means of certain operations on its digits. The meaning of osculation is thus different from that of mainstream mathematics where the term means a contact between curves or surfaces, at which point they have a common tangent. Thus we can speak of osculating circles, meaning two circles that touch at a point through which a common tangent passes. Here is a link to more information on Vedic Mathematics.

Nude Numbers

It was only in my previous post that I mentioned Friedman numbers, named after the former Associate Professor of Mathematics at Stetson University in DeLand, Florida. His name popped up again this morning when I was investigating the number associated with my diurnal age: 26124. Before discussing the mathematical property of this number, namely its "nudity", I'll include some biographical information about Erich Friedman that I found on his website:

My name is Erich Friedman. For 26 years, I was a Professor of Mathematics at Stetson University, located in DeLand, Florida. I retired in 2018 to spend more time on my other interests, including recreational mathematics, puzzles, trivia, and my girlfriend of 30 years. I was born in 1965 in West Lafayette, Indiana. I grew up in Indianapolis and went to North Central High School. I got my bachelor's degree from Rose-Hulman in 1987, and my Ph.D. from Cornell University in 1991, and have been at Stetson ever since.

There's more information on his website but that's enough for this post. Suffice to say that his website looks interesting with many links to mathematical topics. Eric Friedman is the author of OEIS A034838: numbers \(n\) that are divisible by every digit of \(n\). 26214 is a member of this sequence because 1, 2, 4 and 6 do indeed divide into it without remainder.

I was lead to this sequence by a link in Numbers Aplenty concerning what are colorfully called nude numbers, so called because such numbers expose some of their factors. The explanation on the Numbers Aplenty website runs like this:

Y.Katagiri calls a number nude if it is divisible by all of its digits (which should be nonzero) like \(672=6\cdot112=7\cdot96=2\cdot 336\). The number is called "nude" because it exposes some of its factors. There are only \(9039\) such numbers below one million, however there are infinite nude numbers since all repdigits are nude. The smallest nude number which contains all the odd digits is \(1117935\). Note that if a nude number contains a \(5\), then all the other digits must be odd. The smallest nude \(n\) which contains the maximal (8) number of distinct digits is \(1123449768\). The smallest triple of consecutive nontrivial nude numbers is \((1111, 1112, 1113)\). It is easy to see that there cannot be four consecutive nude numbers greater than 10.

The entry goes on to depict the smallest 3 × 3 magic square whose entries are nontrivial consecutive nude numbers (that is, not the numbers from 1 to 9). See Figure 1.

Figure 1

It's easy enough to generate all the nude numbers from 1 to 26124 using the SageMath code shown below (permalink to SageMathCell):

L=[]

for n in [1..26124]:

    N=n.digits()

    OK=1

    for i in range(len(N)):

        if N[i]==0:

            OK=0

            break

        else:

            if n%N[i]!=0:

                OK=0

                break

    if OK==1:

        L.append(n)

print(L)

print("The percentage of nude numbers up to",n,"is",numerical_approx(len(L)/n*100,digits=2))

The output tells us that approximately 2.9% of the numbers between 1 and 26124 are nude. Here is output:

[1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, 124, 126, 128, 132, 135, 144, 155, 162, 168, 175, 184, 212, 216, 222, 224, 244, 248, 264, 288, 312, 315, 324, 333, 336, 366, 384, 396, 412, 424, 432, 444, 448, 488, 515, 555, 612, 624, 636, 648, 666, 672, 728, 735, 777, 784, 816, 824, 848, 864, 888, 936, 999, 1111, 1112, 1113, 1115, 1116, 1122, 1124, 1128, 1131, 1144, 1155, 1164, 1176, 1184, 1197, 1212, 1222, 1224, 1236, 1244, 1248, 1266, 1288, 1296, 1311, 1326, 1332, 1335, 1344, 1362, 1368, 1395, 1412, 1416, 1424, 1444, 1448, 1464, 1488, 1515, 1555, 1575, 1626, 1632, 1644, 1662, 1692, 1715, 1722, 1764, 1771, 1824, 1848, 1888, 1926, 1935, 1944, 1962, 2112, 2122, 2124, 2128, 2136, 2144, 2166, 2184, 2196, 2212, 2222, 2224, 2226, 2232, 2244, 2248, 2262, 2288, 2316, 2322, 2328, 2364, 2412, 2424, 2436, 2444, 2448, 2488, 2616, 2622, 2664, 2688, 2744, 2772, 2824, 2832, 2848, 2888, 2916, 3111, 3126, 3132, 3135, 3144, 3162, 3168, 3171, 3195, 3216, 3222, 3264, 3276, 3288, 3312, 3315, 3324, 3333, 3336, 3339, 3366, 3384, 3393, 3432, 3444, 3492, 3555, 3612, 3624, 3636, 3648, 3666, 3717, 3816, 3864, 3888, 3915, 3924, 3933, 3996, 4112, 4116, 4124, 4128, 4144, 4164, 4172, 4184, 4212, 4224, 4236, 4244, 4248, 4288, 4332, 4344, 4368, 4392, 4412, 4416, 4424, 4444, 4448, 4464, 4488, 4632, 4644, 4824, 4848, 4872, 4888, 4896, 4932, 4968, 5115, 5155, 5355, 5515, 5535, 5555, 5775, 6126, 6132, 6144, 6162, 6168, 6192, 6216, 6222, 6264, 6288, 6312, 6324, 6336, 6366, 6384, 6432, 6444, 6612, 6624, 6636, 6648, 6666, 6696, 6762, 6816, 6864, 6888, 6912, 6966, 6984, 7112, 7119, 7175, 7224, 7266, 7371, 7448, 7476, 7644, 7728, 7777, 7784, 8112, 8128, 8136, 8144, 8184, 8224, 8232, 8248, 8288, 8328, 8424, 8448, 8488, 8496, 8616, 8664, 8688, 8736, 8824, 8832, 8848, 8888, 8928, 9126, 9135, 9144, 9162, 9216, 9288, 9315, 9324, 9333, 9396, 9432, 9612, 9648, 9666, 9864, 9936, 9999, 11111, 11112, 11115, 11122, 11124, 11128, 11133, 11136, 11144, 11155, 11166, 11172, 11184, 11196, 11212, 11222, 11224, 11226, 11232, 11244, 11248, 11262, 11288, 11313, 11316, 11322, 11328, 11331, 11355, 11364, 11412, 11424, 11436, 11444, 11448, 11488, 11515, 11535, 11555, 11616, 11622, 11664, 11676, 11688, 11711, 11824, 11832, 11848, 11872, 11888, 11916, 12112, 12122, 12124, 12126, 12128, 12132, 12144, 12162, 12168, 12184, 12212, 12216, 12222, 12224, 12244, 12248, 12264, 12288, 12312, 12324, 12336, 12366, 12384, 12412, 12424, 12432, 12444, 12448, 12488, 12492, 12612, 12624, 12636, 12648, 12666, 12712, 12726, 12768, 12816, 12824, 12848, 12864, 12888, 12924, 12996, 13113, 13116, 13122, 13128, 13131, 13155, 13164, 13212, 13224, 13236, 13248, 13266, 13272, 13311, 13326, 13332, 13335, 13344, 13362, 13368, 13377, 13392, 13416, 13464, 13488, 13515, 13626, 13632, 13644, 13662, 13713, 13755, 13776, 13797, 13824, 13848, 13896, 13932, 13968, 13995, 14112, 14124, 14128, 14136, 14144, 14184, 14212, 14224, 14232, 14244, 14248, 14288, 14292, 14316, 14328, 14364, 14412, 14424, 14436, 14444, 14448, 14488, 14616, 14664, 14688, 14728, 14784, 14824, 14832, 14848, 14888, 15115, 15135, 15155, 15315, 15515, 15555, 15575, 15715, 16116, 16122, 16128, 16164, 16212, 16224, 16236, 16248, 16266, 16326, 16332, 16344, 16362, 16368, 16416, 16464, 16488, 16626, 16632, 16644, 16662, 16716, 16824, 16848, 16992, 17115, 17122, 17136, 17171, 17199, 17248, 17262, 17444, 17472, 17535, 17717, 17724, 17766, 17955, 18112, 18128, 18144, 18168, 18184, 18216, 18224, 18248, 18264, 18288, 18312, 18336, 18384, 18424, 18432, 18448, 18488, 18624, 18648, 18816, 18824, 18848, 18864, 18872, 18888, 18936, 19116, 19224, 19296, 19332, 19368, 19395, 19692, 19719, 19926, 19935, 19944, 19962, 19971, 21112, 21122, 21124, 21126, 21128, 21132, 21144, 21162, 21168, 21184, 21212, 21216, 21222, 21224, 21244, 21248, 21264, 21288, 21312, 21324, 21336, 21366, 21384, 21412, 21424, 21432, 21444, 21448, 21488, 21492, 21612, 21624, 21636, 21648, 21666, 21672, 21728, 21784, 21816, 21824, 21848, 21864, 21888, 21924, 21996, 22112, 22116, 22122, 22124, 22128, 22144, 22164, 22176, 22184, 22212, 22222, 22224, 22236, 22244, 22248, 22266, 22288, 22326, 22332, 22344, 22362, 22368, 22392, 22412, 22416, 22424, 22444, 22448, 22464, 22488, 22626, 22632, 22644, 22662, 22722, 22764, 22824, 22848, 22888, 22896, 22932, 22968, 23112, 23124, 23136, 23166, 23184, 23226, 23232, 23244, 23262, 23292, 23316, 23322, 23328, 23364, 23412, 23424, 23436, 23448, 23616, 23622, 23664, 23688, 23772, 23832, 23922, 24112, 24124, 24128, 24132, 24144, 24168, 24184, 24192, 24212, 24216, 24224, 24244, 24248, 24264, 24276, 24288, 24312, 24324, 24336, 24384, 24412, 24424, 24432, 24444, 24448, 24472, 24488, 24612, 24624, 24636, 24648, 24696, 24724, 24816, 24824, 24848, 24864, 24888, 24912, 24984, 26112, 26124] 

The percentage of nude numbers up to 26124 is 2.9

Wolfram MathWorld has some auxiliary information:

Numbers in base-10 which are divisible by their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 22, 24, 33, 36, 44, 48, 55, 66, 77, 88, 99, 111, 112, 115, 122, ... (OEIS A034838). Numbers which are divisible by the sum of their digits are called Harshad numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 18, 20, 21, 24, ... (OEIS A005349). Numbers which are divisible by both their digits and the sum of their digits are 1, 2, 3, 4, 5, 6, 7, 8, 9, 12, 24, 36, 48, 111, 112, 126, 132, 135, 144, ... (OEIS A050104). Numbers which are equal to (i.e., not just divisible by) the product of their divisors and the sum of their divisors are called sum-product numbers and are given by 1, 135, 144, ... (OEIS A038369).