Very Special Five Digit Numbers

Analysing the number associated with my diurnal age means that since I turned 10000 days old, those numbers have always contained five digits and will continue to do so for the remainder of my life. Today I turned 27391 days old and that number has a very special quality.

What's obvious at first glance is that all the digits are distinct but less obvious is the fact the absolute values of the differences between successive pairs of digits, which are digits themselves, are also distinct and are different to the digits of the number. As there are four such differences between the five digits of the number, this means that all the digits from 1 to 9 make an appearance.$$ \underbrace{|2-7|}_{5} \, \underbrace{|7-3|}_{4} \, \underbrace{|3-9|}_{6} \, \underbrace{|9-1|}_{8}$$Numbers of this sort belong to OEIS A365257:

A365257

The five digits of a(\(n\)) and their four successive absolute first differences are all distinct.

The OEIS comments state that:

The digit 0 is never present in a(\(n\)) and never appears as a first difference (as this would duplicate in both cases one of the 8 remaining digits involved).

The sequence ends with a(96) = 98274.

The only prime numbers with this property are 39157, 49681, 51869, 53719, 62983, 68749, 68947, 75193, 78259, 89627 and 95287.

The 96 members of this sequence 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

As can be seen, I'm due to experience another such number in a week from today when I reach 27398 days old. My forthcoming 75th birthday, when I am 27394 days old, thus falls between these two special five digit numbers. 

A Truly Incredible Fact About The Number 37

It was this video from the YouTube channel Veritasium that made me aware of the considerable interest attached to the number 37.

I then found this post from a blogger, Chris Grossack, to be especially helpful in explaining the following:

37 is the median value for the second prime factor of an integer; thus the probability that the second prime factor of an integer chosen at random is smaller than 37 is approximately 50%.

He also uses SageMath for his calculations which was an added bonus. Here is the permalink to the calculation to determine the median using the first 100,000 numbers. The output is shown in Figure 1.

Figure 1

The actual proof is summarised in the information contained in Figure 2 which is more than I can comprehend, but I'll include it here:

Figure 2

The blogger uses the formulae shown in Figure 2 to once again show that 37 is the median value. Here is the permalink and the output is shown in Figure 3.

Figure 3

Of course this is not 37's only claim to fame. Wikipedia has an entry for the number 37 and some of the interesting facts contained in that article include a 3 x 3 magic square with 37 at its centre. See Figure 4.

Figure 4

Its magic constant is 37 x 3 = 111, where 3 and 37 are the first and third base-ten unique primes (the second such prime is 11). I wasn't familiar with the notion of a unique prime and so I'll include a definition from the Wikipedia article here:

A prime \(p\) (where \(p\) ≠ 2, 5 when working in base 10) is called unique if there is no other prime \(q\) such that the period length of the decimal expansion of its reciprocal, 1/\(p\), is equal to the period length of the reciprocal of \(q\), 1/\(q\). For example, 3 is the only prime with period 1, 11 is the only prime with period 2, 37 is the only prime with period 3, 101 is the only prime with period 4, so they are unique primes. The next larger unique prime is 9091 with period 10, though the next larger period is 9 (its prime being 333667). Unique primes were described by Samuel Yates in 1980.

I've written about 37 extensively as well in a post titled Star Numbers from the 7th of June 2019. There is a website dedicated to the number 37. It's mentioned by its creator in the YouTube video earlier but, as he himself admits, it hasn't been updated in very many years. However, it still contains a wealth of information.

For example, the site describes a method of determining if a number is divisible by 37. This is the method:

• Divide the number up in groups of three digits, starting from the right.
  (The left-most group may not have three digits.)
• Add the groups together.
• Repeat steps 1 and 2 if the result is still longer than three digits, repeat steps 1 and 2.
• Examine the final three-digit (or smaller) number

The original number is divisible by 37 if and only if this three-digit number is.

I often take note of car number plates here in Jakarta. These are typically of the form B-xxxx where xxxx is a four digit number. It's easy to determine if the four digit number is divisible by 3 because the first digit is simply added to the remaining three. Using leading zeros, the multiples of 37 are:

037, 074, 111, 148, 185, 222, 259, 296, 333, 370, 407, 444, 481, 518, 555, 592, 629, 666, 703, 740, 777, 814, 851, 888, 925, 962, 999

The repeated digit numbers (111 to 999) are a dead given away but the others are not two difficult to identify. Let's consider a number plate like B-1258. The 1258 --> 1 + 258 = 259 = 7 x 37. In this case, the 7 can be divided in to reveal the 37 rather than dealing with division by 37. There is no limit to what can be said about the number 37 but at least in this post and my earlier post of star numbers I've made a start.

Conway's Game of Life Records

Since the 15th February 2024 I've been tracking the number of generations required for the number associated with my diurnal age to reach stability under the rules of Conway's Game of Life. On that date, I created a post titled Diurnal Age Meets Conway's Game Of Life that explained the manner in which this number was arrived at. 

Up until today, the record of around 1190 generations was held by 27373 on the 13th March 2024. At that date, no other number had surpassed 1000 generations. Today however, the number associated with my diurnal age, 27388, exceeded the previous record by an impressive margin. This number required slightly less than 1700 generations to reach stability.

Early in the evolution two gliders were created so these do not appear in the screenshot shown in Figure 1 because by the time stability was reached they were far off screen.

Figure 1: using https://playgameoflife.com/

The path of the gliders can be seen in this alternative view shown in Figure 2 where oscillators appear in black and still life shapes appear as white, both against a background of orange cells that were active prior to stability.

Figure 2: using https://conwaylife.com/

Here's a video of the action:

So the record has been set and it remains to be seen when it will be surpassed but this post formally notes the record and if and when it is exceeded I'll add an addendum.

It's interesting what a difference a single cell that is turned on or off can make. For example, 27389 is identical to 27388 except for one cell that is turned off and thus makes the 8 into a 9. Under Conway's Game of Life rules, it terminates in 142 generations and leaves only two blocks. It's nice how the glider collides with a third block so that the two annihilate each other. Here is a video of the action:

ADDENDUM, Sunday April 14th 2024

27402 stabilises after about 2070 generations under Conway's Game of Life rules to six gliders and an assortment of still lifes and oscillators. This sets the record so far for number of generations. The previous record was held by 27388 with about 1700 generations.

ADDENDUM, Sunday April 28th 2024

Only two weeks since my last addendum and 27419 sets a new record by a significant margin. The new number of generations is about 3745 and Figure 3 shows the final configuration with the paths of the numerous gliders clearly visible.

Figure 3

See later post Another Record in Conway's Game of Life.

Semiprime Runs

Looking at the number associated with my diurnal age today (27386), I noticed that it marked the end of a run of five almost consecutive semiprimes, with only one number that was not a semiprime intervening. Five consecutive semiprimes are not possible because every fourth number must be a multiple of 4 and thus have at least three factors. See Figure 1.

Figure 1

This got me thinking about how common such an occurrence was. As it turns out, not very. In the range up to one million, there are only 211 such numbers (permalink). Up to 40,000, the numbers are:

146, 206, 218, 219, 303, 699, 1142, 1766, 3903, 4538, 6002, 7118, 7863, 9939, 11762, 14258, 16442, 20019, 20283, 22238, 27386, 27519, 27663, 32138, 34418, 35198, 36123, 38163, 38942, 39687

In my SageMath algorithm to find these numbers, I excluded semiprimes that were square numbers but the algorithm is easily modified to include these if necessary (permalink). In the range up to one million, there are 214 such numbers with the smallest being 123:

• \(123 = 3 \times 41 \)
• \(122 = 2 \times 61 \)
• \(121 = 11^2\)
• \(120 = 2^3 \times 3 \times 5\)
• \(119 = 7 \times 17\)
• \(118 = 2 \times 59\)

What about runs of six almost consecutive numbers, that is six semiprimes in a row with only one number that is not a semiprime intervening. These are predictably rather scarce. Excluding semiprimes again and in the range up to one million, there are only seven such numbers and they are (permalink):

219, 143103, 194763, 206139, 273423, 684903, 807663

As can seen, 219 is the first member of this sequence of numbers and its run is as follows:

• \(219 = 3 \times 73\)
• \(218 = 2 \times 109\)
• \(217 = 7 \times 31\)
• \(216 = 2^3 \times 3^3\)
• \(215 = 5 \times 43\)
• \(214 = 2 \times 107\)
• \(213 = 3 \times 71\)

While runs of seven almost consecutive semiprimes should be possible, the SageMath algorithm times out when searching online beyond one million. I have in the past downloaded SageMath to my laptop so that I could search beyond the imposed online limits but my 2013 Macbook Pro seizes up when attempting this.

A Major Milestone

The number associated with my diurnal age today, 27384, has a special property that qualifies it for membership in a rather exclusive OEIS sequence.

A187584

Least number divisible by at least \(n\) of its digits, different and > 1.

Here are members of the sequence for the various values of \(n\):

• \(n =1 \rightarrow 2 =2\)
• \(n =2 \rightarrow 24 = 2^3 \times 3\)
• \(n =3 \rightarrow 248 = 2^3 \times 31\)
• \(n =4 \rightarrow 2364 = 2^2 \times 3 \times 197\)
• \(n =5 \rightarrow 27384 = 2^3 \times 3 \times 7 \times 163\)
• \(n =6 \rightarrow 243768 = 2^3 \times 3 \times 7 \times 1451\)
• \(n =7 \rightarrow 23469768=2^3 \times 3^2 \times 7 \times 46567\)
• \(n =8 \rightarrow 1234759680=2^{12} \times 3^3 \times 5 \times 7 \times 11 \times 29 \)

It can be seen that the final two members of the sequence, 23469768 and 1234759680, have eight and nine digits respectively whereas the earlier members have numbers of digits equal to \(n\). In the case of 27384, the five digits are 2, 3, 4, 7 and 8:

• \( \dfrac{27384}{2} =13692 \)
• \( \dfrac{27384}{3} = 9128\)
• \( \dfrac{27384}{4} = 6846\)
• \( \dfrac{27384}{7} = 3912\)
• \( \dfrac{27384}{8} = 3423\)

There are other OEIS sequences that list all the numbers divisible by at least \(n\) digits and these are:

• a(1)=2=A185186

• a(2)=24=A187516

• a(3)=248=A187398

• a(4)=2364=A187238

• a(5)=27384=A187533

• a(6)=243768=A187534

• a(7)=23469768=A187551

• a(8)=1234759680=A187565

The numbers that are divisible by at least five digits are listed in OEIS A187533 and are:

27384, 29736, 36792, 37296, 37926, 38472, 46872, 73248, 73962, 78624, 79632, 84672, 92736, 123648, 123864, 123984, 124368, 126384, 129384, 132648, 132864, 132984, 134928, 136248, 136824, 138264, 138624, 139248, 139824, 142368, 143928, 146328, 146832, 148392, 148632, 149328, 149832, 162384, 163248, 163824, 164328, 164832, 167328, 167832, 168432, 172368, 183264, 183624, 184392, 184632, 186432, 189432, 192384, 193248, 193824, 194328

27384 is also a Lynch-Bell number. See my blog post titled Lynch-Bell Numbers. 

A Sequence With Only Eight Members

The idea popped into my head to look for numbers that together with their prime factors contain all the digits exactly once. This proved to be a relatively straight forward exercise. Up to one million, there are only eight numbers that qualify. These numbers together with their factorisations are as follows (permalink):

• \(10968 = 2^3 \times 3 \times 457 \)
  
• \(28651 = 7 \times 4093 \)
  
• \(43610 = 2 \times 5 \times 7^2 \times 89 \)
  
• \(48960 = 2^6 \times 3^2 \times 5 \times 17 \)
  
• \(50841 = 3^3 \times 7 \times 269 \)
  
• \(65821 = 7 \times 9403 \)
  
• \(80416 = 2^5 \times 7 \times 359 \)
  
• \(90584 = 2^3 \times 13^2 \times 67 \)

If repeated prime factors are disallowed, then only \(28651\) and \(65821\) qualify. These eight numbers, as I subsequently discovered, make up OEIS A124668:

A124668

Numbers that together with their prime factors contain every digit exactly once.

So this is the sequence with only eight members: 10968, 28651, 43610, 48960, 50841, 65821, 80416, 90584.

Sequence Formed From Digit Display Elements

In my post titled Polyominoes and Conway's Game of Life (February 19th 2024), I looked at the representation of the digits 0, 1 and 2 as polyominoes. In a subsequent post titled Digits 3 to 9 in Conway's Game of Life (February 20th 2024), I examined the digits from 3 to 9 in the same light. Somewhat earlier, in a post titled Diurnal Age Meets Conway's Game Of Life (February 15th 2024), I began to investigate how the number associated with my diurnal age behaves under the Game of Life rules and since 27346 I've been doing this on a daily basis. The results I've been recording in my Airtable database.

My diurnal age today is 27380 and in terms of polyominoes it looks as shown in Figure 1:

Figure 1

This representation uses 54 squares and it occurred to me that starting from 0 and progressing through the natural numbers, records will be set for the number of squares required to represent the numbers. So I set out to determine these record number of squares and the numbers with which they were associated. 

The first step was to set up a data dictionary linking each digit with the number of squares in its polyomino. The dictionary looks like this with digit first followed by the number of squares:

{0:12, 1:5, 2:11, 3:11, 4:8, 5:11, 6:12, 7:7, 8:13, 9:12}

The results in the range from 0 to 100000 are shown in the table in Figure 2 (permalink).

Figure 2

Putting the results in list format, we have the following records:

12, 13, 17, 18, 23, 24, 25, 26, 29, 30, 31, 35, 36, 37, 38, 39, 41, 42, 43, 44, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 59, 60, 61, 62, 63, 64, 65

The numbers associated with these records are:

0, 8, 10, 18, 20, 28, 68, 88, 100, 108, 188, 200, 208, 288, 688, 888, 1000, 1008, 1088, 1888, 2000, 2008, 2088, 2888, 6888, 8888, 10000, 10008, 10088, 10888, 18888, 20000, 20008, 20088, 20888, 28888, 68888, 88888

Surprisingly these numbers make an appearance in OEIS A143617:

A143617

Where record values occur in A010371: number of segments used to represent n on a 7-segment calculator display.

The record values are different since in OEIS A010371 we are counting dashes and not squares. It's the numbers at which these records occur that are the same. The calculator display digits are shown in Figure 3:

Figure 3

Looking at Figure 2 it can be seen that my square total of 54 for today's number of 27380 was reached for the first time way back in 10008. Even though it would be much more labour intensive, another sequence could be developed that counts that number of generations required for a number to reach stability under Conway's Game of Life rules. 

For example, 27380 requires about 380 generations to reach the stable configuration shown in Figures 4 and 5.

Figure 4

Figure 5

The single "toad" and two "traffic lights" alternate between the shapes shown in the two figures whereas the still life "blocks" (two of them), the "pond" (one of them) and the single "honey farm" (the group of four "beehives") remain the same. There's no way of telling how many generations are required for each number to reach stability and so they would all need to be tested individually.

Triple Seven

Triple seven is often associated with a jackpot when it comes up while playing on poker machines so it's interesting to note when this frequency of sevens occurs in numbers associated with my diurnal age. Today I turned 27377 days old:

How many numbers are there, up to one million let's say, with the property that:

• their digits must contain three 7s
• the number itself is divisible by 7
• the other prime factors have digit sums that are divisible by 7

It turns out that there are only 69 such numbers and they are (permalink):

27377, 28777, 67277, 72737, 77357, 77791, 77917, 79177, 154777, 157787, 172277, 177527, 179767, 197477, 227717, 272797, 280777, 287077, 329777, 347767, 367577, 373877, 447727, 448777, 455777, 477673, 507577, 507787, 644777, 677761, 702737, 702877, 706727, 707791, 717227, 717731, 717857, 720377, 726677, 727517, 727783, 732977, 736757, 737387, 737597, 757379, 760277, 764477, 767473, 770273, 771547, 772037, 774557, 776447, 777091, 777203, 777833, 778337, 779107, 782747, 785771, 791077, 827477, 879277, 896777, 917077, 917707, 977137, 977879

The number associated with my diurnal age, 27377, just happens to be the first of them. I won't list the factorisation of all of the above numbers but I will list those up to 100,000:

• \(27377 = 7 \times 3911\)
• \(28777 = 7 \times 4111\)
• \(67277 = 7^2 \times 1373\)
• \(72737 = 7 \times 10391\)
• \(77357 = 7 \times 43 \times 257\)
• \(77791 = 7 \times 11113\)
• \(77917 = 7 \times 11131\)
• \(79177 = 7\times 11311\)

There are variations possible of course. One could simply require that the number contain three 7s and be divisible by 7. In this case, there are 2070 such numbers in the range up to one million with the smallest of them being 777 and the largest of them being 997787 (permalink). 

Alternatively, we look for numbers containing four 7s instead of three. In this case there are only nine such numbers in the range up to one million and they are 772177, 777217, 777721, 777847, 777973, 778477, 784777, 875777 and 977767 (permalink).

While my focus began with the digit 7 and its threefold repetition within a number, it's easy to modify the earlier algorithm so that it tests for the digit 5. In this case the constraints on the numbers are:

• their digits must contain three 5s
• the number itself is divisible by 5
• the other prime factors have digit sums that are divisible by 5

In the range up to one million there are 426 such numbers, the first being 1555 and the last being 998555 (permalink). As with the 7s, the three 5s do no need to be sequential, although they are sequential in these two examples. Their details are as follows:$$ \begin{align} 1555 &= 5 \times 311 \\ 998555 &= 5 \times 41 \times 4871 \end{align}$$We can only test the digits 5 and 7 using this algorithm. The other prime factors cannot have digit sums that are divisible by 2, 3, 4, 6, 8 or 9 because then they would not be prime and the digits 0 and 1 are obviously excluded.

Of Substrings and Divisors

Today I turned 27375 days old and this number has an interesting property in that:$$ \begin{align} 27375 &=375 \times 73  \\ &=5 \times 73 \times 75 \end{align}$$Looking at the numbers on the RHS of the equations, it can be seen that 5, 73, 75 and 375 are all substrings of the string 27375, considering the numbers as collections of characters rather than digits. The numbers with this property form OEIS A059470:

A059470

Numbers that are the products of distinct substrings (>1) of themselves and do not end in 0.

These numbers are not numerous and up to 40000 they are:

125, 375, 735, 1197, 1296, 1352, 1593, 1734, 2346, 3125, 4224, 4872, 5775, 8448, 9072, 11715, 12768, 13455, 14476, 14673, 15625, 16128, 17136, 17493, 18432, 21168, 22176, 23184, 23391, 27216, 27375, 27648, 27864, 32256, 34272, 34398, 36288, 36864, 37296, 39375

The breakdown into divisors/substrings is as follows with some numbers having more than one representation (permalink):

125 equals the product of [25, 5]

375 equals the product of [75, 5]

735 equals the product of [35, 3, 7]

1197 equals the product of [9, 19, 7]

1296 equals the product of [9, 2, 12, 6]

1352 equals the product of [2, 52, 13]

1593 equals the product of [9, 3, 59]

1734 equals the product of [17, 34, 3]

2346 equals the product of [3, 34, 23]

3125 equals the product of [25, 125]

4224 equals the product of [24, 2, 4, 22]

4872 equals the product of [87, 7, 8]

4872 equals the product of [2, 4, 87, 7]

5775 equals the product of [75, 77]

8448 equals the product of [48, 4, 44]

9072 equals the product of [72, 9, 2, 7]

11715 equals the product of [11, 15, 71]

12768 equals the product of [2, 7, 12, 76]

13455 equals the product of [13, 3, 345]

14476 equals the product of [7, 44, 47]

14673 equals the product of [3, 73, 67]

15625 equals the product of [625, 25]

16128 equals the product of [6, 8, 12, 28]

17136 equals the product of [3, 6, 7, 136]

17493 equals the product of [17, 3, 49, 7]

18432 equals the product of [32, 4, 18, 8]

21168 equals the product of [21, 6, 168]

22176 equals the product of [176, 21, 6]

23184 equals the product of [3, 4, 84, 23]

23391 equals the product of [339, 3, 23]

27216 equals the product of [6, 21, 216]

27216 equals the product of [2, 7, 72, 27]

27375 equals the product of [375, 73]

27375 equals the product of [5, 73, 75]

27648 equals the product of [64, 2, 8, 27]

27864 equals the product of [2, 6, 86, 27]

32256 equals the product of [32, 3, 6, 56]

34272 equals the product of [3, 42, 272]

34272 equals the product of [2, 34, 7, 72]

34272 equals the product of [2, 34, 3, 4, 42]

34398 equals the product of [9, 98, 39]

36288 equals the product of [2, 3, 36, 6, 28]

36864 equals the product of [64, 3, 4, 6, 8]

37296 equals the product of [2, 37, 7, 72]

37296 equals the product of [2, 7, 296, 9]

37296 equals the product of [3, 6, 7, 296]

39375 equals the product of [3, 5, 7, 375]

I adapted the code for generating the substrings from this source (see Figure 1).

While these numbers are not frequent, there are two coming up in the relatively near future (27648 and 27864) before there is a big gap to the next number, 32256.

Forming Equations from the Digits of a Number

There was a post that I made to my Pedagogical Posturing blog in August of 2013 before I created this mathematical blog in the second half of 2015. The title was "Forming Equations from Integer Sequences" and that title was perhaps a little misleading. At the time, I wasn't aware of the OEIS or Online Encyclopedia of Integer Sequences. What I meant was the sequence of digits that define a number. For example, today my diurnal age is 27374 and so the sequence of digits is 2, 7, 3, 7, 4. Concatenation of the digits is not allowed. Thus we cannot have 27, 3, 7, 4 for example. 

In that long ago post, I wrote:

Recently I've been using Twitter to create a daily tweet that records my "day count" (number of days I've been alive) plus its factors (if not prime) and some interesting facts about the number itself or one of its factors. Sometimes there's little to say about the number and in such cases I've found that I can usually form an equation by inserting mathematical operators between one or more of the digits. 

For example, yesterday the count was 23518 and 23 - 5 = 18. Today the count is 23519 and 2 + 3 + 5 - 1 = 9. I was wondering if it's always possible to create an equation from five digits using the standard mathematical operators (addition, subtraction, multiplication, division and exponentiation in combination with brackets). Obviously with just two digits, it's only possible when the digits are repeated e.g. 99 becomes 9=9. With three digits, it's sometimes possible e.g. 819 becomes 8 + 1 = 9 but generally it isn't e.g. 219. With four digits, it's more likely e.g. 2119 becomes -2 + 11 = 9 but I'm doubtful whether this is always so. There must come a point however, where the number of digits is sufficient to ensure that it's always so. Maybe five digits is that point.

From now on, I'll try each day to form an equation to test out this theory. For example, tomorrow the count is 23520 which becomes 2 + 3 - 5 = 2 x 0 and it works for tomorrow but beyond that let's see.

Needless to say I didn't "try each day to form an equation to test out this theory" but it might be time to give it a go. The 27374 of my diurnal age today is an easy one:$$2 \times 7 -3=7+4$$However, yesterday's number, 27373, doesn't prove so easy. It seems that having two 3's and two 7's in the number makes things difficult. The conditions that I imposed in the original blog post were the use of only the standard mathematical operators of addition, subtraction, multiplication, division and exponentiation in combination with brackets. These operators needed to be applied to the digits in the order in which they occurred.

One modification that I will make here is to allow \(x \, | \,y\) meaning \(x\) is divided into \(y\) as opposed to \(x/y\) meaning \(x\) is divided by \(y\). This seems quite reasonable as it still only involves the operation of division but allows more flexibility. Its application doesn't seem to help in the case of 27373. If we allow the operator \(x\) // \(y\) meaning return the whole number part of the dividend, then an equation is possible:$$2|(7-3)=7//3$$If we allow // then we could allow \(x\) % \( y\) meaning return the remainder as a whole number when \(x\) is divided by \(y\). Thus 7 % 3 = 1.  This might be termed modulo division.

I can't see any way to create an equation from 27373 without extending the original conditions. Even concatenation of the digits doesn't seem to help. A common symbol for concatenation is || and thus 2 || 7 = 27. However, I've stipulated that the digits are to be treated as separate so I'll adhere to that condition. The best approach is to stick to the original conditions and if a solution is not possible, then // and % can be resorted to.

In the case of 27374, there is more than one way to create an equation. Here is another way:$$ 2 \times 7 -(3+7)=4 $$So what I'll try to do is to reassert my original goal of trying each day to form an equation and see what patterns emerge.

This activity of forming a "digit equation" is not all that different from one of Quanta's mathematical games called "Hyperjumps". See Figure 1.

Figure 1

Looking Back at History

Before I started this Mathematics blog in the second half of 2015, I had started making some mathematical posts in my Pedagogical Posturing blog. I only remembered this when I was looking back over posts that I had made to that blog which I had begun in March of 2009. The first mathematical post I made was on August 5th of 2013 and was titled Reflections on 23500.

The main point that emerged from this post was that, while rounding off to the nearest thousand is common enough, it's also the case that rounding off to the nearest multiple of 500 is also popular, especially when the numbers involved are not large. This is why the number popped up in reference to the populations of various towns and cities, both currently and historically.

The other interesting fact about this number was that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source). Thus:$$\text{radius of Earth }: \text{ radius of Earth's orbit }: : 1:23500$$Of course, this again is an approximation or rounding off of the more accurate figure of 23,460 earth radii or \(1.496 \times 10^{11}\) metres (termed an Astronomical Unit or AU). 

Because of my reliance on the OEIS, I often look at a number in terms of its membership in various sequences but many numbers, like 23500, have a standalone physical significance. Here is the full post made almost 11 years ago now:

Today I'm 23500 days old and I was looking around to see if there was any significance to this number. It's factors are unremarkable, \(2^2 \times 5^3 \times 47 \), but the fact that it's halfway between 23000 and 24000 means that it pops up quite frequently in Internet searches, as would 22500 or 24500 I would imagine. A search reveals that the approximate population of Boston in 1620 was 23500 and there are several towns around the world that are listed as having this population currently e.g. Bishopbriggs in Scotland.

Bishopbriggs grew from a small rural village on the old road from Glasgow to Kirkintilloch and Stirling during the 19th century, eventually growing to incorporate the adjacent villages of Auchinairn, Cadder, Jellyhill and Mavis Valley. It currently has a population of approximately 23,500 people.

Source: http://en.wikipedia.org/wiki/Bishopbriggs

It turns out that Mount Isa has the same population (source):

Mount Isa is located just 200 kilometres from the Northern Territory border and 1,829 kilometres from Queensland’s capital, Brisbane. The nearest major city, Townsville, can be found 883 kilometres from The Isa. Mount Isa covers an area of over 43,310 square kilometres, making it geographically the second largest city in Australia to Kalgoorlie-Boulder, Western Australia ... With a population of approximately 23,500, Mount Isa is a major service centre for north-west Queensland.

Many other examples of towns having populations of about 23500 could be quoted. In addition of populations, the number sometimes comes up as a dollar figure (source):

In its third annual funding cycle, the Black Philanthropy Initiative has pumped $23,500 back into the Winston-Salem area to help African Americans improve their parenting skills.

Interestingly, it turns out that the centre of the Sun is about 23500 times more distant from us than the centre of the Earth (source).

The sun is far enough away (about 23,500 earth radii) that it took a long time before people knew accurately how far away the sun was. Certainly the ancient Greeks had calculated the distance, but they also knew that their results could be off. 

Many countries in the world have five digit postal codes or zip codes as they are sometimes known. These codes identify particular locations within the country e.g. Muang Prachinburi, Prachinburi, Thailand has a postcode of 23500. The United States uses a five digit system but apparently there is no location corresponding to 23500, although there is for 23499 and 23501.

At the time I made this post, I hadn't discovered the OEIS although I was aware of it in late 2014 or early 2015. Nor had I discovered Numbers Aplenty and several other resources that have proved most helpful in finding out more about numbers. 

More Sequences Involving SOD and POD

The terms SOD and POD are used here to refer to Sum Of Digits and Product Of Digits. I've made a post titled SOD ET AL on June 29th 2021. Quite recently on March 10th 2024, I made another post titled Permutations Involving Sum and Product of Digits and like that post, this post involves a combination of SOD and POD.

On March 2nd 2024, I turned 27362 days old and the number 27362 has the following property as noted in my Airtable record:

27362 is a number \(n\) without the digit 0 with two distinct prime factors such that \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) both have two distinct prime factors. Here SOD stands for sum of digits and POD for product of digits. Note that this is different to the arithmetic and multiplicative digital roots of a number. Here the results for \(n\), \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) are: $$27362 = 2 \times 13681\\27382 = 2 \times 13691\\27866 = 2 \times 13933$$The members of this sequence from 27362 up to 40000 are (permalink):

27362, 27373, 27389, 27395, 27419, 27443, 27493, 27515, 27535, 27571, 27578, 27598, 27635, 27641, 27649, 27757, 27842, 27849, 27899, 27933, 27934, 28141, 28187, 28235, 28293, 28321, 28345, 28369, 28498, 28529, 28769, 28783, 28811, 28846, 28874, 28963, 29219, 29227, 29263, 29278, 29291, 29335, 29377, 29485, 29487, 29534, 29543, 29553, 29593, 29594, 29617, 29626, 29657, 29765, 29773, 29797, 29951, 31187, 31273, 31435, 31439, 31462, 31618, 31619, 31631, 31677, 31693, 31754, 31762, 31767, 31783, 31826, 31874, 31893, 32161, 32177, 32179, 32221, 32449, 32521, 32527, 32534, 32551, 32629, 32666, 32735, 32755, 32819, 32827, 32845, 32863, 32881, 33121, 33133, 33431, 33458, 33499, 33523, 33526, 33643, 33658, 33659, 33671, 33729, 33837, 33842, 33877, 33926, 33947, 33963, 33983, 34315, 34321, 34363, 34467, 34514, 34531, 34555, 34634, 34733, 34754, 34829, 34837, 34873, 34966, 34973, 34993, 35138, 35218, 35219, 35233, 35318, 35366, 35414, 35477, 35522, 35611, 35614, 35633, 35657, 35678, 35693, 35726, 35761, 35782, 35789, 35813, 35857, 35887, 35927, 36111, 36154, 36169, 36178, 36193, 36227, 36289, 36398, 36447, 36463, 36485, 36535, 36577, 36641, 36733, 36759, 36853, 36893, 36961, 37165, 37239, 37381, 37486, 37586, 37615, 37678, 37787, 37837, 37865, 37943, 37981, 38137, 38179, 38243, 38245, 38297, 38359, 38422, 38429, 38463, 38473, 38489, 38515, 38549, 38615, 38758, 38771, 38837, 38849, 38854, 38914, 38926, 38957, 38978, 38983, 38999, 39127, 39145, 39257, 39413, 39453, 39481, 39637, 39661, 39723, 39747, 39811, 39871, 39917, 39941, 39959

We can extend this idea to sphenic numbers and consider numbers \(n\) without the digit 0 with three distinct prime factors such that \(n\) + SOD(\(n\)) and \(n\) + POD(\(n\)) both have three distinct prime factors. An example of such a number is 27544 where \(n\), \(n\) + SOD and \(n\) + POD factorise respectively as follows:$$ \begin{align} 27554 &= 2 \times 23 \times 599\\27577 &= 11 \times 23 \times 109\\28954 &= 2 \times 31 \times 467 \end{align}$$The numbers satisfying this condition from 27544 up to 40000 are:

27554, 27671, 27745, 27813, 27914, 27982, 28118, 28217, 28226, 28326, 28353, 28355, 28366, 28514, 28535, 28713, 28819, 28878, 28954, 29559, 29589, 29622, 29829, 29926, 29955, 29958, 31215, 31274, 31538, 31611, 31623, 31642, 31659, 31726, 31742, 31983, 32151, 32195, 32218, 32241, 32326, 32394, 32421, 32457, 32542, 32631, 32739, 32829, 32862, 32883, 32997, 33226, 33297, 33319, 33341, 33438, 33454, 33586, 33734, 33765, 33882, 33971, 34131, 34143, 34359, 34498, 34539, 34622, 34655, 34683, 34773, 34941, 34953, 34959, 34977, 35165, 35185, 35265, 35371, 35529, 35686, 35866, 35949, 36177, 36249, 36381, 36417, 36534, 36597, 36669, 36698, 36718, 36743, 36933, 37118, 37222, 37247, 37262, 37378, 37383, 37497, 37522, 37542, 38234, 38253, 38337, 38355, 38361, 38395, 38566, 38674, 39219, 39238, 39242, 39263, 39277, 39282, 39369, 39462, 39515, 39538, 39542, 39621, 39639

These sequences do not appear in the OEIS and I certainly won't be submitting them (pearls before swine) but they are interesting examples of sequences arising from a combination of SOD and POD.

27372: Another Palindromic Day

Days like today, when I turn 27372 days old, pop up every one hundred days during the course of a millennium of days and there is a 110 day gap between millennia. So, for example, from 27972 to 28082, there will be a gap of 110 days. Today's number shares some important properties with another palindrome, 26362, that I created a post about on June 6th 2021. It was titled 26362: Another Special Palindrome. 

One property that the two share is that they are both members of OEIS  A070001:

A070001

Palindromic integers > 0, whose 'Reverse and Add!' trajectory (presumably) does not lead to another palindrome.

Up to 40000, the members of this sequence are not numerous and they are:

4994, 8778, 9999, 11811, 19591, 22822, 23532, 23632, 23932, 24542, 24742, 24842, 24942, 26362, 27372, 29792, 29892, 33933, 34543, 34743, 34943, 39493

It can be seen that 26362 and 27372 are consecutive and 1010 days apart in terms of my diurnal age. As I wrote in the post previously alluded to:

These palindromes are not regarded as potential Lychrel numbers because they are already palindromes and some of them are the result or end point of \(k\) + reverse(\(k\)) iterations. However, some are not and these, I think, deserve special consideration. These are:

19591, 23532, 23932, 24542, 24742, 24942, 26362, 27372, 29792, 33933, 34543, 34743, 34943, 39493

So 26362 and 27372 are paired again and they are only the 7th and 8th palindromes to have the simultaneous property that:

• they cannot be derived from \(k\) + reverse(\(k\)) for one or more values of \(k\)
• their Reverse and Add trajectories (presumably) do not lead to another palindrome 

These two numbers are also members of OEIS A045960:

A045960

Palindromic even lucky numbers.

Up to 40000, the initial members are:

2, 4, 6, 22, 44, 212, 262, 282, 434, 474, 646, 666, 818, 838, 868, 2442, 2662, 2772, 4884, 4994, 6666, 6886, 8118, 8338, 20202, 20402, 21012, 21812, 22322, 22422, 22922, 23332, 23532, 24042, 25652, 26162, 26262, 26562, 26762, 27372, 28682

A property that 27372 doesn't share with 26762 is that the former's arithmetic digital root is the same of its middle digit. Of the three and five digit palindromes in the range up to 40000, there are only 36 that satisfy this condition. They are (permalink):

919, 929, 939, 949, 959, 969, 979, 989, 999, 18181, 18281, 18381, 18481, 18581, 18681, 18781, 18881, 18981, 27172, 27272, 27372, 27472, 27572, 27672, 27772, 27872, 27972, 36163, 36263, 36363, 36463, 36563, 36663, 36763, 36863, 36963

For example, the arithmetic digital root of 27372 is 2 + 7 + 3 + 7 + 2 = 21 and 2 + 1 = 3. The middle digit of 27372 is 3.

Permutations Involving Sum and Product of Digits

One sequence that I was surprised NOT to find in the OEIS was one that involves adding the sum and product of a number's digits to the number itself and then comparing the two results. If the results are different but one is a permutation of the other, then the original number is a member of the sequence. Obviously numbers containing the digit 0 will not qualify as the product of the digits will always be 0. This first number to qualify is 36 where we have:

• sum of digits = 3 + 6 = 9
• number + sum of digits = 45
• product of digits = 3 x 6 = 18
• number + product of digits = 54
• 45 and 54 are permutations of the digits 4 and 5

In the range up to 40,000, there are 80 such numbers and they are (permalink):

36, 156, 438, 1145, 3228, 3348, 3414, 3711, 4314, 4689, 5769, 5949, 6219, 7311, 8343, 9216, 11245, 11257, 11439, 11523, 11558, 11619, 12145, 12512, 12821, 13266, 13512, 14346, 14512, 15123, 15212, 15312, 15412, 15512, 15612, 15712, 15812, 16119, 16236, 16344, 16512, 17512, 18221, 18484, 18512, 18551, 18844, 21145, 21512, 21699, 21821, 22314, 23214, 23238, 24216, 24574, 25112, 25474, 27237, 27369, 27999, 28121, 28233, 29331, 31266, 31512, 31896, 32214, 32238, 33597, 34299, 34461, 34554, 34632, 34776, 35112, 35445, 36216, 37341, 38232

For most of these numbers, the two results of adding the sum and the product of the digits to the number produce permutations with digits that are not identical to the original number. However, there are three numbers where this is indeed the case and these numbers are 5769, 14346 and 27369 (permalink):

• 5769 --> 5796 and 7659
• 14346 --> 14364 and 14634
• 27369 --> 27396 and 29637

These numbers are listed in the OEIS and form the initial members of OEIS A246421:

A246421

Numbers \(n\) such that (\(n\) + digit sum of \(n\)) and (\(n\) + digit product of \(n\)) are nontrivial permutations of the digits of \(n\).

All the digit sums and the digit products are multiples of 9. The first members of the sequence are as follows:

5769, 14346, 27369, 41346, 52569, 56925, 94725, 122346, 126135, 129213, 143658, 152469, 154269, 155169, 157914, 162135, 192213, 212346, 216135, 219213, 221346, 236124, 238959, 245925, 261135, 263124, 291213, 326124, 328536, 344925, 361647, 362124, 367425, 368892, 392436, 413658

I would surmise that such a series is finite because as the numbers get larger the size of the product of digits when added to the original number generates numbers with far more digits. There are two associated OEIS sequences to OEIS A246421 and they are:

A246420

Numbers \(m\) such that (\(m\) + digit sum of \(m\)) is a permutation of the decimal digits of \(m\).

Listed below are some of numbers coming up for me in terms of my diurnal age (with 27369 marking the starting point):

27369, 27513, 27558, 27702, 27747, 27891, 27936, 28035, 28224, 28269, 28413, 28458, 28602, 28647, 28836, 29124, 29169, 29313, 29358, 29502, 29547, 29736, 29925, 30123, 30168, 30312, 30357, 30501, 30546, 30735, 30924, 30969, 31023, 31068, 31212, 31257, 31401, 31446, 31635, 31824, 31869, 32112, 32157, 32301, 32346, 32535, 32724, 32769, 32913, 32958, 33012, 33057, 33201, 33246, 33435, 33624, 33669, 33813, 33858, 34101, 34146, 34335, 34524, 34569, 34713, 34758, 34902, 34947, 35001, 35046, 35091, 35235, 35424, 35469, 35613, 35658, 35802, 35847, 36135, 36324, 36369, 36513, 36558, 36702, 36747, 36891, 36936, 37035, 37224, 37269, 37413, 37458, 37602, 37647, 37836, 38124, 38169, 38313, 38358, 38502, 38547, 38736, 38925, 39024, 39069, 39213, 39258, 39402, 39447, 39636, 39780, 39825

A243102

Numbers \(n\) such that the digits of (\(n\) + product of digits of \(n\)) are a nontrivial permutation of the digits of \(n\).

Listed below are some of numbers coming up for me in terms of my diurnal age (with 27369 marking the starting point):

27369, 28179, 28195, 29123, 29154, 29213, 29381, 29397, 29873, 31126, 31213, 31235, 31238, 31259, 31354, 31365, 31561, 31925, 32113, 32265, 32286, 32341, 32352, 32492, 32538, 32743, 32793, 33125, 33129, 33142, 33158, 33186, 33248, 33253, 33294, 33455, 33456, 33475, 33558, 33585, 33965, 33967, 34135, 34156, 34167, 34351, 34356, 34526, 34535, 34553, 34563, 34599, 34655, 34951, 35123, 35134, 35165, 35231, 35262, 35267, 35361, 35463, 35616, 35625, 35652, 35673, 35684, 35763, 35794, 35837, 35861, 35974, 36123, 36154, 36178, 36213, 36381, 36722, 36825, 36935, 37168, 37813, 37849, 38143, 38153, 39183, 39251