Power Towers

The number associated with my diurnal age today, \( \textbf{28144}\), has the property that its power tower of digits is equal to its product of digits. That is to say:

2 ^ 8 ^ 1 ^ 4 ^ 4 = 2 x 8 x 1 x 4 x 4 = 36

Such numbers are relatively rare. In the range beyond 28144 up to 40000, there are only the following numbers that satisfy: 28182, 31111, 33113 and 33131.

Meanwhile, I thought it was time to clarify the reasons behind the right associativity of power towers and so I consulated Gemini. Here was the response:

Let's break down the mechanics of power towers, tetration, and the crucial role of associativity in how we evaluate these operations.

Associativity: The Order of Operations

In mathematics, associativity dictates how we group numbers when an expression contains multiple instances of the same operator.

• Left-Associative: Operations are evaluated from left to right. Subtraction and division are left-associative by convention.
  For example, 10 − 5 − 2 is evaluated as (10 − 5) − 2 = 3.
• Right-Associative: Operations are evaluated from right to left.

Exponentiation is mathematically defined as right-associative. When you see a sequence of exponents, you evaluate them from the top down.

Here is why this convention exists: if exponentiation were left-associative, it would be mathematically redundant. Let's look at 2³².

• If it were left-associative: (2³)² = 8² = 64.
  By the rules of exponents, (a^b)^c = a^{b × c}. Therefore, a left-associative power tower is just a disguised way of multiplying exponents together (2^{3 × 2} = 2⁶ = 64). We already have multiplication, so we don't need a new convention for this.
• Because it is right-associative: 2⁽³²⁾ = 2⁹ = 512.
  Evaluating from the top-down creates a fundamentally new, uniquely powerful operation that cannot be simplified by basic multiplication.

Power Towers

A power tower is simply a visual stack of exponents, written as a^bcd.

Because of the right-associative rule, you always start at the highest point of the tower and work your way down to the base.

For example, to evaluate 2³²²:

1. Start at the top: 2² = 4
2. Move down one level: 3⁴ = 81
3. Calculate the final base: 2⁸¹

The result is 2,417,851,639,229,258,349,412,352.

Tetration: Iterated Exponentiation

Tetration is a specific type of power tower where the base number is identical throughout the entire stack.

Just as multiplication is repeated addition, and exponentiation is repeated multiplication, tetration is repeated exponentiation. It is the fourth fundamental hyperoperation in mathematics (following addition, multiplication, and exponentiation).

Because standard exponentiation notation becomes visually cumbersome when stacked high, mathematicians use specific notations for tetration:

• Knuth's Up-Arrow Notation: a ↑↑ n
• Rudy Rucker's Superscript Prefix: ⁿa

In both notations, a is the base and n is the height of the tower.

Examples of Tetration:

• ²3 (or 3 ↑↑ 2) = 3³ = 27
• ³3 (or 3 ↑↑ 3) = 3³³ = 3²⁷ = 7,625,597,484,987

The Computational Horizon

Tetration generates exponential growth at a scale that rapidly breaks the limits of standard computation.

If we take just one more step up the tower to ⁴3 (or 3 ↑↑ 4), the expression becomes 3³³³, which evaluates to 3^{7,625,597,484,987}.

This number has over 3.6 trillion decimal digits. While theoretical mathematics can easily define and manipulate the expression ⁴3, calculating the exact integer quickly leaves the realm of standard datatypes. Numbers at this scale become highly unrenderable in their full decimal form; storing just the digits of ⁴3 as a standard text file would require terabytes of disk space. When computational number theory investigates numbers at or beyond this threshold, it relies on custom algorithms, modulo arithmetic, or symbolic representation frameworks to analyze the properties of the digits without attempting to hold the entire integer in memory.

Energetic Numbers Revisited

In June of 2024, I posted about Energetic Numbers which are numbers that can be split into two or more parts and these parts, when raised to appropriate integer powers and added together, equal the original number. An example is 27476:$$ 27476 =27^3 + 4^6 + 7^4 + 6^4 $$In that post I made reference to a text file that listed the first 10000 energetic numbers but did not show how the numbers could be split and recombined.

Fortunately, I was able to get Gemini to create a SageMath program (permalink) that accomplishes this task whenever it is possible. I was prompted to attempt this because 27892, the number associated with my diurnal age today, is one of the numbers listed in the previously mentioned text file. For this number we have:$$27892=27^3 + 8 + 9 + 2^{13}$$Here is a list of some of the upcoming energetic numbers. Notice the run of consecutive numbers from 28260 to 28269 (permalink):

\(27892 = 27^3 + 8 + 9 + 2^{13}\) 
\(27923 = 2^{13} + 7^5 + 9^3 + 2^3 + 3^7\)
\(27962 = 27^3 + 9^2 + 6 + 2^{13}\)
\(27972 = 27^3 + 97 + 2^{13}\)
\(27984 = 2^9 + 7^5 + 9^4 + 8 + 4^6\)
\(28132 = 2^{12} + 8^4 + 1 + 3^9 + 2^8\)
\(28160 = 2^{11} + 8^3 + 160^2\)
\(28203 = 2^9 + 8 + 20^3 + 3^9\)
\(28224 = 28^3 + 2^{11} + 2^7 + 4^6\)
\(28228 = 2^{14} + 82^2 + 2^{10} + 8^4\)
\(28243 = 28^3 + 2^3 + 4^6 + 3^7\)
\(28245 = 2^{13} + 8^3 + 2^5 + 4^7 + 5^5\)
\(28260 = 2^{14} + 8^4 + 2^2 + 6^5 + 0\)
\(28261 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 1\)
\(28262 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 2\)
\(28263 = 2^{14} + 8^4 + 2^2 + 6^5 + 3\)
\(28264 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 4\)
\(28265 = 2^{14 }+ 8^4 + 2^2 + 6^5 + 5\)
\(28266 = 2^{14} + 8^4 + 2^2 + 6^5 + 6\)
\(28267 = 2^{14} + 8^4 + 2^2 + 6^5 + 7\)
\(28268 = 2^{14} + 8^4 + 2^2 + 6^5 + 8\)
\(28269 = 2^{14} + 8^4 + 2^2 + 6^5 + 9\)

27892 also has the property that it can be split into two parts and the two parts, when added together, form a palindromic prime with a prime index:$$ \begin{align} 27892 &\rightarrow 27 + 892 \\  &=919          \\ &=\text{prime}(157) \end{align}$$

Forming Digit Equations: A Game

Source

It occurred to me how forming digit equations from five digits might make for an interesting game. After all, Wordle consists of guessing the five letters that make up a hidden word. In the game I'm conceiving of, a five digit number between 10000 and 99999 would be generated. Let's say the number generated is 25529. The challenge is to use only the following operators to form a digit equation without altering the order of the digits:

+    addition

-    subtraction

x    multiplication

/    divide by

|    divide into

^    exponentiation

(     left bracket

)    right bracket

Figure 1 shows one possible representation:

Figure 1

The same digit equation would look mathematically as follows:$$ \Big ( 2 + \frac{5}{5} \Big )^2=9$$This is somewhat on the difficult side. Obviously an effective game nowadays involves a catchy user interface so coloured balls that pop up to start the game would be effective. The operators shown above would need to be able to be dragged and dropped between the digits in order to generate the equation. The successful creation of an equation would need to be displayed in mathematically readable way such as was done with the example of 25529.

The game could have different levels starting at the fairly elementary level. For example, 10348 could be rendered as:$$1+0+3+4=8$$This equation involves only the a single operation, addition, and the correct placement of the equal sign. By contrast, 25529 is considerably more challenging and involves the use of brackets, addition, division and exponentiation. Of course, for some numbers it is impossible to form an equation within the imposed constraints. An example is 27497 which is unsolvable as far as I can see. This is a long term project but something that I will keep thinking about or perhaps I'll discover that someone has already created such a game. Who knows?

For earlier posts on this theme see my posts titled The Number Plate Game from the 29th June 2024 and Forming Equations from the Digits of a Number on 14th March 2024. To generate some random five digits numbers follow this permalink.

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.

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

Representing Numbers With Digits

My previous post focused on the number 1089 in which I made reference to a newly discovered blogging site at https://math1089.in/ and in particular to a post about the number 1089. In that post it was noted that:$$ \begin{align} 1089 &= 12 \times 3^4 + 5 \times 6 + 78 + 9\\1089 &= 987 + 65 + 4 + 32 + 1 \end{align}$$In another post about the number 108, it was noted that:$$ \begin{align} 108 &= 1 + 2 + 3 + 4 + 5 + 6 + 78 + 9\\108 &= 9 + 8 \times 7 + 6 \times 5 + 4 \times 3 + 2 ‒ 1 \end{align}$$This got me thinking about whether it was possible to represent every number at least once in terms of consecutive single or concatenated digits separated by the basic operations of arithmetic combined with exponentiation and brackets.

For example, my diurnal age today is 27339. Is such a feat possible for this number? The determination is not easy because there are just  so many possible ways to combine the digits from 1 to 9. I spent quite some time playing around with the possibilities and I did come close but not close enough. The exercise is oddly addictive. There's no serious mathematics involved in the exploration. It's more in the nature of a puzzle, like Sudoku. 

There are a variety of strategies, one of which is to establish base points with as few digits as possible. An example of this is:$$35016=1+2+3+4+5+6^7/8+9$$Here the digits 6, 7 and 8 combine to form 34992 and the remaining digits are free to be manipulated. It's true that we can use 34567 to get to a similar number but here five digits are tied up and there's little that can be done with the remaining digits (1, 2 and 8, 9) since they are separated. Here is an example of how we can get close to 27339 using the earlier mentioned base point:$$28084=-(12^3 \times 4+5)+6^7/8+9$$Another approach is to use factors. We know that 27339 = 3 x 13 x 701 and it's easy enough to create the first two factors using 1 + 2 = 3 and 3 x 4 - 5 + 6 = 13. However, we are then left with 789:$$ \begin{align} 27339 &=3 \times 13 \times 701\\30771 &= (1+2) \times (3 \times 4 - 5 +6) \times 789\\ &=3 \times 13 \times 789 \end{align}$$While this works fine for 30771, it's not of much use for 27339. In general, this approach is quite restrictive and the more promising approaches will involve additions and subtractions along with exponentiation, multiplication and division.

At the moment I don't have a solution to the specific problem of representing 27339 in terms of sequential digits and I certainly don't have an answer to the general problem of whether such a representation is always possible or only sometimes possible. Certainly there's an upper limit on the number size and this is imposed by the nine digit restriction but such a limit is huge and my investigation is focusing on numbers in the region of 30000. My suspicion is that it's not always possible within the restrictions imposed. If we relax the requirement that the digits need to be in sequential order or we allows square roots, factorials etc. then maybe it's possible but for the moment I'll keep within the earlier rules and keep revisiting the problem from time to time.

Another OEIS Submission

After tinkering with this sequence for a couple of weeks, I decided to submit it to the OEIS in its original form. I'd tried various ways to make it more interesting but, in the end, it was its inherent simplicity that won out. Here it is:$$a_n=\text{ sum of digits of }\big (a_{n-1} \big )^{a_{n-2}} \text{ where } a_0=1 \text{ and } a_1=2$$I was able to use SageMathCell to generate the terms up to and including \(a_{17}\) but after that the algorithm times out. Here is a permalink with the SageMath code (in blue) and the output (in red) as follows:

a, b=1,2

L=[a, b]
for n in [1..17]:
c=b^a
c=sum(c.digits())
L.append(c)
a, b=b, c
print(L)

[1, 2, 2, 4, 7, 7, 25, 34, 151, 331, 1690, 3265, 26449, 64528, 574513, 1671208, 16090657, 54199564, 559922497] 

Without the sum of digits, this exponentiation would become too large too quickly and the resultant sequence would be fairly meaningless. The sum of digits brings the exponent back to a much lower number but, even so, by \(n=17\) the term is more than half a billion. In my submission, I was able to provide a link to GeeksforGeeks that shows the code to calculate the sum of digits of a given number to a given power in Python3, C++, Java, C# and PHP. Maybe this will give my submission a little more weight. See Figure 1 for a screenshot.

Figure 1

I'm well aware that this is quite a simple sequence but it will have to do until I come up with something like the Collatz or PrimeLatz trajectory sequences. I'll keep trying. Meanwhile, it will be interesting to see how my submission is greeted by the guardians of OEIS who guard their sequences jealously and are suspicious of newcomers. I'll add to this post as news arrives. Here is a link to my previous contribution: OEIS A335789 under the name Sean Lestrange (August 14th 2020). I have an earlier entry, OEIS A301938, under the name Sean Reeves (March 28th 2018). As to the name change, well that's another story.

UPDATE: well, that was quick. When I checked today (16th November 2020), the sequence had already been approved: OEIS A338917. See Figure 2:

Figure 2

Forming Equations from Integers

Read on!

Once upon a time, before I started this exclusively mathematical blog in 2015, I used to post occasional mathematical content to my Pedagogical Posturing blog at https://voodoo-guru.blogspot.com. I was looking back at some of these posts and noticed this one, from Saturday, 24 August 2013, titled Forming Equations from Integer Sequences. It's only short so I'll quote it in full:

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 possible 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 \(2+3-5=2 \times 0 \) and it works for tomorrow but beyond that let's see.

Well, I didn't keep my promise of trying to form an equation each day from the digits making up my diurnal age. I have however, written about selfie numbers in a post of Friday, 27th March 2020 to my Mathematical Meanderings blog site. These are somewhat similar in spirit. In that post, I also mention Friedman numbers that can be described as follows:

Consider \(28547 =(8+5)^4−(7 \times 2)\) expressed in base 10, both sides use the same digits. An integer is a Friedman number if it can be put into an equation such that both sides use the same digits but the right hand side has one or more basic arithmetic operators (addition, subtraction, multiplication, division, exponentiation) interspersed. Brackets, as usual, are essential to clarify the order of operations. These numbers are named after Erich Friedman, Assoc. Professor of Mathematics at Stetson University. With the help of his students he has researched Friedman numbers in bases 2 through 10 and even with Roman numerals. When both sides use the digits in the same order, the number is called a ”nice” or ”strong” Friedman number. For example, \( 3125=(3+[1\times2])^5.\)

My approach is similar to this except I'm trying to create an equation from the digits (using them in the same order as they appear in the number). As another example, today I'm \(26121\) days old and this is an easy one because \( (2 \times 6)/12=1 \) if we allow concatenation of digits. If only individual digits are allowed, then \(-(2-6)=1+2+1 \) satisfies. It's probably better to use the individual digits as is done with the Friedman numbers. I'll try to include this as part of my daily number analysis. 

Going back a few days, we have:

• \(26120 \text{ --> } 2 = \frac{6}{1 + 2 +0} \)
• \(26119 \text{ --> } 2 + 6 \times 1 = -1 + 9 \)
• \(26118 \text{ --> } 2 + 6 + 1 = 1 + 8 \)
• \(26117 \text{ --> } 2 + 6 = 1 \times 1 + 7 \)
• \(26116 \text{ --> } 2 + 6 = 1 + 1 + 6 \)
• \(26115 \text{ --> } -2 + 6 + 1 \times 1 = 5 \)
• \(26114 \text{ --> } -2 + 6 = 1 - 1 + 4 \)
• \(26113 \text{ --> } -2 + 6 = 1 \times 1 + 3 \)
• \(26112 \text{ --> } -2 + 6 = 1 + 1 + 2 \)
• \(26111 \text{ --> } 2 = \frac{6}{1+1+1}\)
• \(26110 \text{ --> } 2 \times 6 \times (1-1) = 0 \)

It seems that it's always possible to form an equation using only brackets and the basic arithmetic operators of addition, subtraction, multiplication, division and exponentiation. Perhaps even the exponentiation is not needed, as the examples above show. Let's see if this assumption holds true for future numbers. This is hardly high level mathematics but it's a simple yet oddly satisfying activity.