The Number Plate Game

Figure 1

Most number plates in Indonesia display a four digit number as shown in Figure 1 where the number is 1234. A game I like to play as I drive around or am driven around is to try to make a digit equation from these four digits. My rules are that only the four operations of addition, subtraction, multiplication and division are allowed together with exponentiation and brackets. Concatenation of digits is not allowed. Division can be of two types:

• a digit divided into the next digit e.g. 2 | 4 = 2 which reads 2 divided into 4
• a digit divided by the next digit e.g. 4 / 2 = 2 which reads 4 divided by 2

For the 1234 in Figure 1, the digit equation is very straightforward:$$ \begin{align}  -1 + 2 &= -3 + 4 \\ 1 &=1 \end{align}$$Theoretically there are \(10^4 = 10000 \) possible numbers ranging from 0000 to 9999. Any numbers with at least two zeroes will form a digit equation. Take 1009 as an example:$$ \begin{align} 1 \times 0 &= 0 \times 9 \\ 0 &=0 \end{align} $$If the digit 1 appears at least twice in the first three digits, then a digit equation is always possible. Take 1814 as an example:$$\begin{align} 1^8 &= 1^4 \\ 1 &=1 \end{align}$$Two consecutive digits are often helpful because they can be collapsed to a 1. Take 6599 as an example:$$\begin{align} 6 - 5 &= 9 \, /  \,9 \\ 1 &=1 \end{align}$$One or more zeroes are also helpful because raising any digit to the zero power produces a 1. Take 9032 as an example:$$\begin{align} 9^0 &= 3-2 \\ 1 &=1 \end{align}$$Brackets also prove very useful as shown in the example of 4894 where we bracket the 8 and 9 and make the 8 negative:$$\begin{align} 4^{-8+9} &= 4 \\ 4^1 &=4 \\ 4&=4  \end{align}$$Of course, forming a digit equation can require a little trickery. Take 3649 as an example:$$\begin{align} 3^{ 6-4}&= 9 \\ 3^2 &=9 \\ 9 &= 9 \end{align}$$Sometimes a digit equation is not possible according to the rules imposed. Take 3637 as an example. I can't see how to make a digit equation out of those four digits.

On the other hand, it's often possible to form digit equations in several different ways. Take our original 1234 as an example where another representations is possible:$$\begin{align} 1^2 &= -3+4 \\ 1 &=1 \end{align}$$It would be an interesting exercise to work out digit equations for all the numbers between 0000 and 9999 and to make a note of those numbers where an equation is not possible. In fact I've began this process in an Airtable database (link) and it would appear that 0124 is the first four digit number to thwart the creation of a digit equation. I'll keep working on the database and make additions to this post when I discover numbers of interest. It will of course take some time to complete.

See post titled Forming Equations from the Digits of a Number from the 14th of March 2024.

ADDENDUM: July 10th 2024

A variation of this game would be to see if one can get the numbers to total zero. This is very similar to the game already described and every solution to that game becomes a solution to this game because \(a=b \implies a-b=0\). This new, let's call it "zero game", allows for additional numbers plates containing the digit zero to qualify. For example 2470:$$2470 \rightarrow 2 \times 4 \times 7 \times 0 = 0$$Under the old rules this doesn't qualify but here the product of the digits is zero so it does. Another example would be 6772:$$6772 \rightarrow 6 \times  (7 - 7) \times 2 = 0$$Under the old rules, a digit equation could not be formed from these digits but here it can.

Unity as a Sum of Egyptian Fractions

Continuing with Achmad Damar's "104 Number Theory", I found this little problem interesting. The author asks: let \(k\) be an even number. Is it possible to write 1 as the sum of the reciprocals of \(k\) odd integers? He approaches the problem by assuming that:$$ 1=\frac{1}{n_1}+ \cdots + \frac{1}{n_k}$$for odd integers \(n_1, \dots, n_k \). Removing denominators produces:$$n_1 \cdots n_k=s_1+ \cdots + s_k $$where all \(s_i\) are odd because numbers that are the products of odd numbers are themselves odd. This is impossible because the LHS is odd and the RHS is even because there are an even number (\(k\)) of integers and the sum of each pair of odd  integers is even. Thus it is not possible to represent 1 as the sum of the reciprocals of an even number of odd integers. However, if \(k\) is odd, then it is possible. The example is given of unity expressed as the sums of reciprocals of nine odd integers. See Figure 1.

Figure 1

This got me thinking about how this result was obtained. There are 114 odd integers between 3 and 231 inclusive and 7,032,112,662,630 ways to sample 9 reciprocals at a time (that's over seven trillion ways). Nonetheless, when running this program in SageMathCell, the above combination of fractions was quickly spat out and after that the program timed out. Running the same program in my jupyter notebook (to avert the program timing out), no further combinations were generated. Is this combination of reciprocals unique? I'm not sure.

Certainly we can state that there are infinitely many disjoint sets of positive integers in which the sum of those reciprocals is equal to unity (source). I have a program that generates Egyptian fractions for rational numbers \(p/q\) where \(p<q\). Using this program, I can determine that: $$ \begin{align} \frac{5}{7} &= \frac{1}{2}+\frac{1}{5}+\frac{1}{70}\\ \frac{2}{7} &= \frac{1}{4}+\frac{1}{48}\\1 &= \frac{1}{2}+\frac{1}{4}+\frac{1}{5}+\frac{1}{48}+\frac{1}{70} \end{align}$$Alternatively, I could take another pair of fractions that add to 1 such as 7/11 and 4/11. This gives:$$ \begin{align} \frac{7}{11} &= \frac{1}{2} + \frac{1}{8} + \frac{1}{88} \\ \frac{4}{11} &= \frac{1}{3} + \frac{1}{33} \\ 1 &= \frac{1}{2}+ \frac{1}{3}+ \frac{1}{8} + \frac{1}{33} + \frac{1}{88} \end{align} $$Obviously we could continue this process indefinitely. This Mathematics Stack Exchange source states that:

R. L. Graham showed that \( a(n)>0 \) for \( n>77 \), where \( a(n) \) is the number of ways to express 1 as the sum of distinct unit fractions such that the sum of the denominators is \( n \). This implies that for values of \( n \leq 77 \) such representations may not be possible. In the two examples shown earlier the sum of denominators is greater than 77. However, representations where \(n \leq 77 \) are certainly possible. For example from the same Stack Exchange source we see that:$$ 2+3+11+22+33 = 2+5+8+12+20+24 \text{ and both total }71\\2+4+9+12+18 =2+5+6+12+20 \text{ and both total }45$$and the sum of reciprocals of LHS's and RHS's all total 1.

The simplest representation of 1 using Egyptian fractions is:$$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$$but once we impose special conditions such as all denominators must be odd or prime or whatever, then things get more complicated. This site offers some useful insights into a problem that can well be explored in far more depth than I've attempted here.

Two Simple Number Properties

Reading through Achmad Damar's "104 Number Theory", there are two simple number properties that are mentioned early on that are easy to prove but the proofs are elegant to my eye at least. Let's look at them.

PROPERTY ONE

Let \(n\) be an integer greater than 1. Prove that \(2^n\) is the sum of two consecutive odd integers. 

Let's suppose that the statement is true and that:$$ \begin{align} 2^n &= (2k-1)+(2k+1)\\ \implies \, 2^n &= 4k \\ k &= 2^{n-2}\\ \text{thus } 2^n &= (2^{n-1}-1) + (2^{n-1}+1) \end{align}$$From this we can see how to quickly calculate the two odd consecutive numbers. Suppose \(n=10\) and thus \(2^{10}=1024\). Because \(2^9=512\), it's easy to see that \(1024 = 511 + 513\).

PROPERTY TWO

Let \(n\) be an integer greater than 1. Prove that \(3^n\) is the sum of three consecutive integers.

Again, let's suppose that the statement is true and that:$$ \begin{align} 3^n &= (s-1)+s+(s+1) \\ \implies \, 3^n &= 3s\\ s &= 3^{n-1} \\ \text{thus } 3^n &= (3^{n-1}-1 )+ 3^n + (3^{n-1}+1) \end{align} $$Again it's easy to find these three numbers. Let's take \(n=3\) so that \(3^3=27\) and \(3^2=9\). This gives \(27=8+9+10\).

Energetic Numbers

I was surprised today to stumble upon a new category of number called energetic numbers. Such number are reasonably common: the 10,000th such number is 103,718. These sorts of numbers form OEIS A055480: 

A055480

Energetic numbers: numbers that can be broken into two or more substrings and expressed as a sum of (possibly different) positive powers of those substrings.

The examples are given of \(142 = 14^1 + 2^7\) and \(8833 = 88^2 + 33^2\). This property is reminiscent of d-powerful numbers but with these only the individual digits can be used. For example, 27472 can be expressed as:$$27472= 2^3 + 7^4 + 4^3 + 7^5 + 2^{13}$$My diurnal age today is 27476 and, while it is not a d-powerful number, it is an energetic number because it can be expressed as follows:$$27476=27^3+4^6+7^4+6^4$$So it is almost a d-powerful number but not quite. The initial energetic numbers are:

24, 43, 63, 89, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 132, 135, 142, 153, 175, 209, 224, 226, 262, 264, 267, 283, 284, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 568, 598, 629, 739, 794, 809, 849, 935, 994, 1000

However, the OEIS reference has a link to a text file that lists the first 10,000 such numbers. Unfortunately, the file does not list the representation of the number in terms of powers of substrings. For 27472, I had to simply experiment until I found the right combination. The text file contains the C program code used to generate the list. I used Google's Gemini to convert the code to Python but it wouldn't generate any output using SageMathCell.

The energetic numbers will contain the d-powerful numbers as a subset. The initial d-powerful numbers are:

24, 43, 63, 89, 132, 135, 153, 175, 209, 224, 226, 262, 264, 267, 283, 332, 333, 334, 357, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 407, 445, 463, 518, 598, 629, 739, 794, 849, 935, 994

Comparing this sequence to OEIS A055480, it can be seen that 100 is the first energetic number that is not d-powerful because:$$ \begin{align} 100 &\neq 1^2 + 0^2 + 0^2 \text{ whereas}\\ 100 &=10^2 + 0^2 \end{align}$$Similarly for the numbers from 101 to 109. Here is a list of the energetic numbers between 27476 and 40000:

27476, 27479, 27493, 27496, 27497, 27498, 27527, 27529, 27536, 27549, 27562, 27568, 27569, 27617, 27635, 27639, 27663, 27697, 27720, 27736, 27747, 27749, 27752, 27764, 27765, 27790, 27817, 27823, 27856, 27892, 27923, 27962, 27972, 27984, 28132, 28160, 28203, 28224, 28228, 28243, 28245, 28260, 28261, 28262, 28263, 28264, 28265, 28266, 28267, 28268, 28269, 28288, 28306, 28332, 28336, 28355, 28375, 28395, 28403, 28423, 28425, 28449, 28513, 28519, 28532, 28533, 28534, 28536, 28539, 28553, 28554, 28569, 28574, 28593, 28599, 28603, 28607, 28613, 28682, 28730, 28731, 28732, 28733, 28734, 28735, 28736, 28737, 28738, 28739, 28819, 28932, 28937, 28955, 29130, 29232, 29253, 29259, 29263, 29287, 29319, 29324, 29342, 29343, 29346, 29347, 29349, 29369, 29370, 29385, 29435, 29436, 29444, 29454, 29474, 29532, 29543, 29586, 29634, 29637, 29732, 29734, 29744, 29755, 29765, 29769, 29835, 29853, 29923, 29943, 29947, 29967, 30135, 30153, 30175, 30236, 30243, 30257, 30289, 30312, 30340, 30341, 30342, 30343, 30344, 30345, 30346, 30347, 30348, 30349, 30373, 30375, 30427, 30445, 30600, 30601, 30602, 30603, 30604, 30605, 30606, 30607, 30608, 30609, 30628, 30935, 30964, 31032, 31096, 31122, 31131, 31169, 31216, 31234, 31237, 31296, 31314, 31324, 31339, 31342, 31346, 31347, 31362, 31366, 31376, 31385, 31393, 31452, 31455, 31549, 31558, 31563, 31636, 31673, 31695, 31852, 31920, 31921, 31922, 31923, 31924, 31925, 31926, 31927, 31928, 31929, 31932, 32052, 32072, 32097, 32214, 32234, 32235, 32252, 32254, 32256, 32275, 32296, 32297, 32325, 32326, 32327, 32342, 32355, 32364, 32367, 32368, 32436, 32437, 32492, 32524, 32525, 32527, 32528, 32548, 32562, 32565, 32587, 32588, 32595, 32635, 32637, 32655, 32672, 32722, 32731, 32743, 32749, 32780, 32781, 32782, 32783, 32784, 32785, 32786, 32787, 32788, 32789, 32792, 32793, 32800, 32801, 32802, 32803, 32804, 32805, 32806, 32807, 32808, 32809, 32810, 32811, 32812, 32813, 32814, 32815, 32816, 32817, 32818, 32819, 32820, 32821, 32822, 32823, 32824, 32825, 32826, 32827, 32828, 32829, 32830, 32831, 32832, 32833, 32834, 32835, 32836, 32837, 32838, 32839, 32840, 32841, 32842, 32843, 32844, 32845, 32846, 32847, 32848, 32849, 32850, 32851, 32852, 32853, 32854, 32855, 32856, 32857, 32858, 32859, 32860, 32861, 32862, 32863, 32864, 32865, 32866, 32867, 32868, 32869, 32870, 32871, 32872, 32873, 32874, 32875, 32876, 32877, 32878, 32879, 32880, 32881, 32882, 32883, 32884, 32885, 32886, 32887, 32888, 32889, 32890, 32891, 32892, 32893, 32894, 32895, 32896, 32897, 32898, 32899, 32903, 32905, 32907, 32908, 32913, 32914, 32917, 32918, 32922, 32923, 32924, 32925, 32926, 32927, 32928, 32930, 32931, 32932, 32933, 32934, 32935, 32936, 32937, 32938, 32939, 32940, 32941, 32942, 32943, 32944, 32945, 32946, 32947, 32948, 32949, 32952, 32953, 32954, 32963, 32965, 32968, 32969, 32973, 32978, 32979, 32980, 32981, 32982, 32983, 32984, 32985, 32986, 32987, 32988, 32989, 32994, 32995, 32997, 32998, 33020, 33021, 33022, 33023, 33024, 33025, 33026, 33027, 33028, 33029, 33042, 33068, 33078, 33084, 33087, 33102, 33106, 33123, 33125, 33134, 33158, 33162, 33165, 33182, 33219, 33220, 33221, 33222, 33223, 33224, 33225, 33226, 33227, 33228, 33229, 33236, 33242, 33246, 33247, 33260, 33261, 33262, 33263, 33264, 33265, 33266, 33267, 33268, 33269, 33272, 33273, 33274, 33276, 33282, 33283, 33284, 33286, 33288, 33292, 33298, 33318, 33322, 33338, 33372, 33387, 33398, 33413, 33422, 33427, 33428, 33429, 33442, 33447, 33467, 33485, 33486, 33512, 33532, 33538, 33539, 33548, 33562, 33568, 33578, 33582, 33592, 33622, 33623, 33624, 33642, 33648, 33658, 33677, 33685, 33686, 33752, 33757, 33772, 33775, 33778, 33779, 33786, 33792, 33798, 33804, 33822, 33823, 33824, 33826, 33828, 33829, 33830, 33834, 33842, 33843, 33844, 33846, 33847, 33864, 33865, 33872, 33873, 33874, 33878, 33882, 33884, 33893, 33942, 33952, 33954, 33957, 33974, 33987, 33992, 34036, 34038, 34112, 34118, 34131, 34146, 34162, 34186, 34217, 34226, 34230, 34231, 34232, 34233, 34234, 34235, 34236, 34237, 34238, 34239, 34244, 34254, 34256, 34265, 34268, 34272, 34274, 34276, 34279, 34290, 34291, 34292, 34293, 34294, 34295, 34296, 34297, 34298, 34299, 34322, 34323, 34326, 34327, 34342, 34366, 34367, 34374, 34377, 34387, 34388, 34402, 34423, 34439, 34472, 34474, 34498, 34525, 34526, 34528, 34529, 34542, 34562, 34568, 34578, 34638, 34652, 34672, 34674, 34677, 34689, 34698, 34727, 34728, 34739, 34746, 34762, 34766, 34770, 34771, 34772, 34773, 34774, 34775, 34776, 34777, 34778, 34779, 34784, 34786, 34832, 34834, 34836, 34852, 34869, 34872, 34873, 34874, 34892, 34922, 34924, 34925, 34928, 34947, 34948, 34980, 34981, 34982, 34983, 34984, 34985, 34986, 34987, 34988, 34989, 35022, 35024, 35032, 35080, 35081, 35082, 35083, 35084, 35085, 35086, 35087, 35088, 35089, 35178, 35203, 35205, 35208, 35213, 35216, 35220, 35221, 35222, 35223, 35224, 35225, 35226, 35227, 35228, 35229, 35231, 35233, 35243, 35245, 35247, 35248, 35274, 35284, 35285, 35287, 35289, 35323, 35332, 35352, 35358, 35372, 35378, 35427, 35447, 35487, 35488, 35522, 35552, 35598, 35625, 35628, 35630, 35662, 35688, 35738, 35809, 35825, 35845, 35848, 35885, 35887, 35912, 35930, 35931, 35932, 35933, 35934, 35935, 35936, 35937, 35938, 35939, 35978, 35979, 35992, 35994, 36104, 36114, 36122, 36144, 36152, 36158, 36164, 36232, 36234, 36237, 36239, 36253, 36254, 36258, 36274, 36275, 36277, 36278, 36283, 36292, 36294, 36296, 36314, 36324, 36327, 36332, 36342, 36343, 36346, 36348, 36368, 36382, 36385, 36408, 36412, 36417, 36435, 36437, 36438, 36454, 36457, 36472, 36473, 36474, 36478, 36492, 36497, 36522, 36544, 36547, 36558, 36567, 36568, 36637, 36656, 36657, 36693, 36708, 36719, 36722, 36723, 36724, 36731, 36733, 36742, 36743, 36746, 36748, 36762, 36775, 36789, 36837, 36854, 36877, 36892, 36922, 36924, 36942, 36980, 36981, 36982, 36983, 36984, 36985, 36986, 36987, 36988, 36989, 37002, 37045, 37072, 37083, 37102, 37177, 37215, 37216, 37220, 37221, 37222, 37223, 37224, 37225, 37226, 37227, 37228, 37229, 37242, 37243, 37244, 37246, 37248, 37262, 37263, 37264, 37265, 37267, 37268, 37282, 37283, 37284, 37306, 37332, 37333, 37336, 37348, 37352, 37353, 37358, 37372, 37392, 37393, 37410, 37412, 37420, 37421, 37422, 37423, 37424, 37425, 37426, 37427, 37428, 37429, 37442, 37443, 37449, 37455, 37462, 37482, 37483, 37484, 37485, 37486, 37578, 37626, 37642, 37645, 37648, 37663, 37687, 37702, 37714, 37732, 37738, 37747, 37794, 37826, 37827, 37843, 37851, 37867, 37884, 37953, 37997, 38124, 38133, 38145, 38164, 38235, 38257, 38272, 38317, 38322, 38324, 38325, 38326, 38343, 38345, 38348, 38349, 38372, 38382, 38383, 38394, 38456, 38474, 38514, 38521, 38522, 38527, 38560, 38563, 38566, 38576, 38655, 38657, 38746, 38762, 38834, 38835, 38923, 38924, 39142, 39276, 39338, 39340, 39341, 39342, 39343, 39344, 39345, 39346, 39347, 39348, 39349, 39352, 39354, 39368, 39372, 39373, 39384, 39392, 39393, 39402, 39420, 39421, 39422, 39423, 39424, 39425, 39426, 39427, 39428, 39429, 39432, 39442, 39443, 39446, 39462, 39469, 39472, 39482, 39483, 39484, 39485, 39532, 39533, 39536, 39538, 39608, 39623, 39624, 39627, 39628, 39629, 39634, 39642, 39648, 39662, 39682, 39687, 39689, 39734, 39749, 39752, 39758, 39822, 39823, 39825, 39827, 39834, 39842, 39844, 39848, 39864, 39882, 39884, 39886

As with the energetic numbers 100 to 109 inclusive, runs of consecutive integers are common e.g. 30600, 30601, 30602, 30603, 30604, 30605, 30606, 30607, 30608, 30609 where we have:$$30600 = 30^3+60^2+0$$and the digits 1 to 9 can be substituted for the zero and the numbers remain energetic.

Pythagorean Triangles With Integer Sides

It's not obvious from the description in the OEIS that the numbers forming the sequence can in fact be with associated with Pythagorean triads and right angled triangles. Today I turned 27475 days old and one of the properties of this number is its membership of OEIS A334542.

A334542

Numbers \(m\) such that \(m^2 = p^2 + k^2\), with \(p\) > 0, where \(p\) = A007954 (\(m\)) = the product of digits of \(m\).

27475 has a product of digits equal to 1960. Let's form a right angles triangle with 27475 as the hypotenuse and 1960 as one of the two sides forming the right angle. Let's call the other side \(x\). The relationship between \(x\), 1960 and 27475 can be expressed as:$$ \begin{align} x^2+1960^2 &= 27475^2\\x^2 &= 27475^2-1960^2\\x &= \sqrt{27475^2-1960^2} \\  &= 27405 \end{align} $$The square root and value of \(x\) just happens to be an integer but this is rarely the case. In general, with \(m\) as the number and \(p\) as its product of digits (with \(p\) > 0), we will have:$$ \begin{align} x^2 &= m^2-p^2 \\ x &= \sqrt{m^2-p^2} \end{align} $$ Here is a permalink to a program that will generate the 16 members of OEIS A334542 sequence up 40000, excluding the trivial single digit numbers, and showing the Pythagorean triads as well. With the Side 1 column being the products of digits and the Hypotenuse column being the sequence numbers, the results are:

Thus the members of the sequence are 58, 85, 375, 666, 1968, 1998, 3578, 3665, 3891, 4658, 4995, 6675, 7735, 18434, 27475 and 28784. Note the appearance of the Number of the Beast, 666, in the sequence.

The same approach can be made using the sum of digits instead of the product of digits. The only results are 17, 25 and 85 with the familiar triads shown below (Side 1 shows the sum of digits --> permalink):

These types of Pythagorean triads have the property that they are self-referencing. The number and its sum of digits or product of the digits form two sides and Pythagoras' Theorem takes care of the remaining side.

As another example, let's investigate whether there are numbers such that the number and its reversal form a Pythagorean Triad. It turns out that there are eight such numbers in the range up to 100,000. Ignoring the numbers formed by the reversals, these are 56, 5265, 5656, 12705, 56056, 55517, 51557 and 59248 and they form the triads (permalink) shown below (Side 2 column shows the numbers and the Hypotenuse column shows their reversals):

What's Special About Palindrome 27472?

Palindromic numbers occur every century during a millenium and the millenium I'm focused on stretches from 27000 to 27999. Because the first two digits of numbers in this millenium add to 9, the palindromes that arise have the peculiarity that the middle digit is always the arithmetic digital root, with the exception of 27072. Thus we have:

27172 with digital root of 1

27272 with digital root of 2

27372 with digital root of 3

27472 with digital root of 4

27572 with digital root of 5

27672 with digital root of 6

27772 with digital root of 7 

27872 with digital root of 8 

27972 with digital root of 9

I'll soon be 27472 days old and I've gotten into the habit of creating a post for each palindromic day. So what other special properties does this palindrome have? 

• It is a palindrome in base 9 as well $$27472_{10} \rightarrow 41614_{ \, 9}$$This qualifies it for membership in OEIS A180454: numbers that are 5-digit palindromes in at least two bases.
  
  
• It is a d-powerful number, because it can be written as $$27472=2^3 + 7^4 + 4^3 + 7^5 + 2^{13} $$
• It can be written as a sum of two squares in two different ways because it is a product of a power of 2 and two 4k+1 primes:$$ \begin{align} 27472 &= 2^4 \times 17 \times 101\\ &=24^2+ 164^2\\ &=56^2+ 156^2 \end{align}$$
•  27472 is an untouchable number, because it is not equal to the sum of proper divisors of any number.
  
  
• 27472 is a palindrome with exactly six prime factors (counted with multiplicity) and this qualifies it for membership in OEIS A046332 whose members, up to 40000, are:
  
  
  2772, 2992, 6776, 8008, 21112, 21712, 21912, 23632, 23832, 25452, 25752, 25952, 27472, 28782, 29392
  
  
• 27472 is a palindrome which is even and in which the parity of digits alternates. This qualifies it for membership in OEIS A030149. The initial members are:
  
  
  0, 2, 4, 6, 8, 212, 232, 252, 272, 292, 414, 434, 454, 474, 494, 616, 636, 656, 676, 696, 818, 838, 858, 878, 898, 21012, 21212, 21412, 21612, 21812, 23032, 23232, 23432, 23632, 23832, 25052, 25252, 25452, 25652, 25852, 27072, 27272, 27472
  
  
• The Collatz Trajectory for 27472 is:

27472, 13736, 6868, 3434, 1717, 5152, 2576, 1288, 644, 322, 161, 484, 242, 121, 364, 182, 91, 274, 137, 412, 206, 103, 310, 155, 466, 233, 700, 350, 175, 526, 263, 790, 395, 1186, 593, 1780, 890, 445, 1336, 668, 334, 167, 502, 251, 754, 377, 1132, 566, 283, 850, 425, 1276, 638, 319, 958, 479, 1438, 719, 2158, 1079, 3238, 1619, 4858, 2429, 7288, 3644, 1822, 911, 2734, 1367, 4102, 2051, 6154, 3077, 9232, 4616, 2308, 1154, 577, 1732, 866, 433, 1300, 650, 325, 976, 488, 244, 122, 61, 184, 92, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1

There are 108 steps required to reach 1. Figure 1 shows a plot of the numbers using a logarithmic scale for the vertical axis. 

 

Figure 1

• The Aliquot Sequence for 27472 is:

27472, 29444, 25240, 31640, 50440, 73040, 114448, 117680, 156112, 174224, 163366, 121862, 81418, 40712, 46648, 61352, 53698, 26852, 28210, 36302, 25954, 15086, 8794, 4400, 7132, 5356, 4836, 7708, 6404, 4810, 4766, 2386, 1196, 1156, 993, 335, 73, 1, 0

            Figure 2 shows a plot of these numbers. 

Figure 2

• The Anti-Divisors of 27472 are:

3, 5, 7, 9, 11, 15, 27, 32, 33, 37, 45, 47, 55, 99, 111, 135, 165, 167, 185, 297, 329, 333, 407, 495, 544, 555, 999, 1169, 1221, 1485, 1665, 2035, 3232, 3663, 4995, 6105, 7849, 10989, 18315 

• The Arithmetic Derivative of 27472 is 56832
  

• The Maximum - Minimum Recursive Algorithm for 27472 produces:

27472, 54945, 50985, 92961, 86922, 75933, 63954, 61974, 82962, 75933 

• The Minimal Goldbach decomposition of 27472 is 23 and 27449
  

• The number of steps required is to reach home prime is 5 :
• 27472
• 222217101
• 333310925169
• 3365956198099
• 1111271910230901
• 3419034730977487
  

• The multiplicative persistence of 27472 is as follows: 27472, 784, 224, 16, 6
  
  
• 27472 is a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (28458). There are many such partitions, one example of which is:
  
  [8, 34, 136, 808, 27472] and [1, 2, 4, 16, 17, 68, 101, 202, 272, 404, 1616, 1717, 3434, 6868, 13736] both of which sum to 28458
  
  The divisors of 27472 are 1, 2, 4, 8, 16, 17, 34, 68, 101, 136, 202, 272, 404, 808, 1616, 1717, 3434, 6868, 13736 and 27472.
  

• 27472 has Odds and Evens Trajectory of length 1 and is 27472, 27478, 27478
  
  
• 27472 is a pseudoperfect number because it is the sum of a subset of its proper divisors. There are many such subsets one of which is [101, 1616, 1717, 3434, 6868, 13736]. Permalink.
  
  
• 27472 is a practical number, because each smaller number is the sum of distinct divisors of 27472.  For example, take an arbitrary number like 23891. It can expressed as the sum of distinct divisors of 24742 in several different ways e.g. the sum of 1, 17, 68, 272, 404, 808, 1717, 6868 and 13736.
  
  
• 27472 is of course an abundant number, since it is smaller than the sum of its proper divisors (29444).

27472 has the property, shared by all the numbers in its decade, that its digit sum is given by the concatenation of its first and last digit, here 22. Thus:

• 27470 has digit sum 20 
• 27471 has digit sum 21 
• 27472 has digit sum 22
• 27473 has digit sum 23 
• 27474 has digit sum 24 
• 27475 has digit sum 25 
• 27476 has digit sum 26 
• 27477 has digit sum 27 
• 27478 has digit sum 28 
• 27479 has digit sum 29

This permalink will generate a list of all 1230 numbers in the range up to 40000. However, there are only 13 palindromes with this property and they are 191, 2992, 10901, 11711, 12521, 13331, 14141, 25852, 26662, 27472, 28282, 29092 and 39993.

Under Conway's Game of Life rules, the 27472 shape shown at the beginning of this blog stabilises after about 914 generations to the shapes shown in Figures 3 and 4 with the paths of the five gliders visible in Figure 4.

Figure 3

Figure 4

Remarkable Reversals

Consider the following:$$27456=2^6 \times 3 \times 11 \times 13\\65472=31 \times 11 \times 3 \times 2^6$$As can be seen, if we reverse the order of the digits of 27456, then the resultant number (65472) has, as its prime factors, all the prime factors of 27456 but reversed. Admittedly, 2, 3 and 11 are palindromic but its still a remarkable result, made even more so by the fact that both numbers (27456 and 65472) have exactly nine prime factors (with multiplicity).

Ignoring the reversal itself then, up to one million, there is only one other number with these properties and that is 238656:$$238656 = 2^6 \times 3 \times 11 \times 113\\656832 = 311 \times 11 \times  3 \times 2^6$$However, if we remove the requirement that the prime factors themselves must be reversed and require only that the number and its reverse both have nine prime factors with multiplicity then we have the following numbers up to one million (again only the smaller number and not its larger reversal is included):

21168, 23424, 23616, 27456, 41184, 212256, 213192, 215232, 219072, 230208, 236925, 236928, 238656, 251505, 251748, 253824, 255024, 257856, 259968, 271728, 276696, 276768, 291168, 293328, 299808, 373464, 403056, 403488, 404064, 422208, 424116, 424764, 428928, 441045, 441288, 462384, 472608, 492048, 606096, 610688, 612576, 635328, 804168

These numbers form OEIS A109029 (permalink):

A109029

Numbers that have exactly nine prime factors counted with multiplicity (A046312) whose digit reversal is different and also has 9 prime factors (with multiplicity).

The algorithm to generate these numbers is quite flexible and if we change the requirement to eight prime factors with multiplicity then, up to one million, we get 147 numbers as opposed to 43 for nine factors. If we change the requirement to ten prime factors then only 11 numbers satisfy and these are 46848, 217152, 219456, 232848, 257664, 259776, 274104, 276048, 415584, 428736, 846369. For 11 factors we only have two numbers satisfying: 295245 and 426816 while for 12 prime factors there are none.

A Special Fourth Power Diophantine Equation

Consider the Diophantine equation:$$a^4+b^4+c^4+d^4=(a+b+c+d)^4$$It's clear that no set of positive numbers will satisfy this equation, at least one of the numbers will need to be negative. It's also clear that if we find a solution \(n=a+b+c+d\) then this will generate an infinity of solutions between all multiples of this number will satisfy the equation:$$ \begin{align} (na)^4+(nb)^4+(nc)^4+(nd)^4 &= (n *(a+b+c+d))^4\\n^4  a^4 +n^4  b^4 + n^4  c^4 + n^4  d^4 &= n^4  (a+b+c+d)^4 \\a^4+b^4+c^4+d^4 &= (a+b+c+d)^4 \end{align} $$where \(n\) can be any integer, positive or negative, or even zero. So what is the first number that satisfies the equation? In 1964, somebody named Brudno found the number 5491 that satisfies where  5491 is written as 955 + 1700 - 2364 + 5400 and we have:$$955^4+1700^4+(-2364)^4+5400^4\\=(955 + 1700 - 2364 + 5400)^4$$In terms of positive numbers, this means that the progressive multiples of 5491 are also solutions viz. 10982, 16473, 21964, 27455, 32946, 38437, 43928, 49419, ... etc..

Are there other numbers? Well somebody named Wroblewski found 51361 to be a solution when it written as 48150 - 31764 + 27385 + 7590. Thus:$$48150^4 +(-31764)^4+27385^$+7590^4\\=(48150 - 31764 + 27385 + 7590)^4$$At this point, it's relevant to include this quote from the comments to OEIS A138760: numbers \(n\) such that \(n^4\) is a sum of 4th powers of four nonzero integers whose sum is \(n\):

Any multiple of a member is also a member. A member that is not a multiple of another member is called primitive. Using elliptic curves, Jacobi and Madden prove that there are infinitely many primitive members. According to them, the only primitive members less than 222,000 are 5491 (due to Brudno) and 51361 (due to Wroblewski).

The comments then list a number greater than 220,000 and that number is 1347505009. The reason I came across these numbers is that today my diurnal age is 27455 and that is the fifth multiple of 5491.

World of Numbers

Today I came across an interesting website via a link in an OEIS entry for 27454, the number associated with my diurnal age as of today's date. Figure 1 shows a screenshot.

Figure 1

The link provided to P. De Geest's  Nine Digits Digressions takes us to a particular page on the World of Numbers website. Figure 2 shows the page and Figure 3 shows the home page of the website.

Figure 2

Figure 3

Looking at the website, I immediately thought that it was one of those websites that had been created in the 1990s and then abandoned. However, a closer look showed that it had been created in 1996 but updated on June 2nd 2024 which is the date on which I'm creating this post. So remarkably the site has been maintained from 1996 to 2024 by P. De Geest.

So who is P. De Geest? Well his site provides a not-so-recent photo and a brief bio:

Photo taken in 2004

E-mail: pdg@worldofnumbers.com 

Web Page: http://www.worldofnumbers.com/index.html 

My name is Patrick De Geest, born on the 9th of October 1956, in Wezembeek-Oppem, Belgium (about 10 km east of Brussels), unmarried, mildly myopic, graduated in architecture but never practiced the profession. Currently I'm an employee working in the aircargo export sector (National Airport Zaventem). I didn't lose my interest in beautiful patterns and proportions though, and managed to transfer it to the field of numbers. 

Also, through the years, I gradually became familiar with the use of personal computers (no, I'll never sell my first Sinclair ZX81) and learned for programming techniques (basic, assembly, ...). All these 'creativities' culminated recently in a website about recreational mathematics with 'palindromes' as the main topic. I opted for palindromes not because of my length (181 cm), my average weight (77 kg) or my housenumber (141) but because I was attracted by their overall symmetry and the fact that it was a novel and thus insufficiently studied subject. Thanks to many contributors from all over the world the site is still expanding. 

For the rest I'm a rather quiet individual who likes to read an occasional book, watch a movie, listen to classical music, travel once or twice a year to a near/far exotic  destination and bike from time to time when the weather permits.

Anyway the point is that the site contains a wealth of information about curious number properties with Patrick giving the following overview of the site's contents:

In this well-filled website you'll find a multitude of facts and figures about topics from the  World!Of Numbers . Don't look for a logical order. It is an amalgamation of randomly gathered numbers, curios, puzzles, palindromes, primes, gems, your much valued contributions and more general information. Enjoy! Patrick De Geest  

Like Taneja's papers described in my previous post, there is great content here for future posts to this blog. Getting back to the original OEIS sequence, we see that:

\(27454^ {0.25} = 12.\overline{87215934}68573\)

The first nine digits of the decimal part do indeed contain all the digits from 1 to 9. Interestingly I can find no reference to these sorts of calculations of page 7 of "Nine Digits" topic. Perhaps it's on one of the other pages. Numbers like 27454 are part of OEIS A034279:

A034279

Decimal part of \(a(n)^{1/4}\) starts with a 'nine digits' anagram.

The sequence begins: 7396, 8751, 8933, 8950, 9070, 11184, 26484, 26522, 27454, 30858, 36923, 39895, 40828, 42793, 47311, 58738, 58985, 61143, 72788, 73506, 75636, 79562, 80138, 80260, 81101, 83261, 94796, 96256, 101915, 102189, 103310, 103416, 108901

There's no reason to restrict ourselves to the fourth root and there are sequences corresponding to numbers raised to  1/2, 1/3, 1/5, 1/6, 1/7 and 1/8 powers and probably more. Here is a permalink to a general purpose algorithm that will generate sequences for any power desired. The relevant OEIS sequences are:

• square root: OEIS A034277 with initial members being 86, 868, 1278, 5211, 7494, 7772, 14567, 17573, 18421, 20844, 24960, 26535, 29172, 29301, 29987, 32845
  
  
• cube root: OEIS A034278 with initial members being 429, 939, 7015, 11456, 15221, 17521, 21000, 21160, 22397, 24789, 28916, 30945, 33743, 35440, 36732
  
  
• fifth root: OEIS A034280 with initial members being 12, 1635, 2112, 6905, 15376, 18660, 18795, 20085, 21086, 21447, 22064, 23077, 23540, 25817, 27040, 28204, 30668, 31258, 31287, 37407, 38533
  
  
• sixth root: OEIS A034281 with initial members being 648, 695, 1979, 7509, 9214, 12567, 19740, 21555, 24235, 24646, 25624, 27427, 30717, 30748
  
  
• seventh root: OEIS A034282 with initial members being 551, 574, 2998, 8265, 9407, 10357, 12459, 15885, 20480, 26103, 26134, 29297, 35096, 35984, 37113, 39084, 39733, 39735
  
  
• eighth root: OEIS A034283 with initial members being 3927, 4176, 10041, 10827, 13575, 15544, 15853, 17244, 20154, 24759, 25146, 30008, 30038, 30635, 30692, 32046, 37215

That's enough I think. Remember that there are factorial 9 ways to arrange the nine digits and this equals 362880, an impressive number of permutations.