Midpoint of the Year

On the 7th August 2021, I made a post title Key Points in the Year and Figure 1 is an extract from that post:

Figure 1

In that post I failed to mention that the exact midpoint of a common or non-leap year occurs at noon on the 2nd of July. As today is June 30th, that point is now not far off. In leap years, the midpoint occurs at midnight on the 2nd of July. This middle day can be written as \( \textbf{20260702} \) using YYYYMMDD format or \( \textbf{2026183} \) using the year and the number of days that have elapsed in it. 

Both formats uniquely define any date but I prefer the latter for the purposes of number analysis. Unfortunately, feeding both of these numbers into my daily number analysis program caused the wheels to spin and the analysis didn't complete. The program handles the five digit numbers associated with my diurnal age but seven and eight digit numbers are too much for my M1 Macbook Air's processor to handle.

However, 2026183 is prime and its reciprocal has a period of 2026182. It forms a twin prime pair with 2026181. It's interesting to consider how many times 183 leads to primes when concatenated with a range of years from 2000 to 2100. Here are the years:

2005, 2006, 2011, 2014, 2015, 2018, 2021, 2023, 2026, 2030, 2041, 2047, 2069, 2093, 2098

Of these, only 2005, 2026 and 2047 form twin prime pairs when both 181 and 183 are concatenated with them.

Counting the Days

On this final day of July 2024, it seems appropriate to say something about the date which can be written in YYYYMMDD format as 20240731. In a post from the 26th of January 2023 titled Turning Dates into Numbers, I looked at 2023 in the light of this number format and so I should do the same for 2024 now that we are more than halfway through it. I have an algorithm that I created for generating the list of days in 2024 which, being a leap year, will contain 366 entries. Here is the list:

20240101, 20240102, 20240103, 20240104, 20240105, 20240106, 20240107, 20240108, 20240109, 20240110, 20240111, 20240112, 20240113, 20240114, 20240115, 20240116, 20240117, 20240118, 20240119, 20240120, 20240121, 20240122, 20240123, 20240124, 20240125, 20240126, 20240127, 20240128, 20240129, 20240130, 20240131, 20240201, 20240202, 20240203, 20240204, 20240205, 20240206, 20240207, 20240208, 20240209, 20240210, 20240211, 20240212, 20240213, 20240214, 20240215, 20240216, 20240217, 20240218, 20240219, 20240220, 20240221, 20240222, 20240223, 20240224, 20240225, 20240226, 20240227, 20240228, 20240229, 20240301, 20240302, 20240303, 20240304, 20240305, 20240306, 20240307, 20240308, 20240309, 20240310, 20240311, 20240312, 20240313, 20240314, 20240315, 20240316, 20240317, 20240318, 20240319, 20240320, 20240321, 20240322, 20240323, 20240324, 20240325, 20240326, 20240327, 20240328, 20240329, 20240330, 20240331, 20240401, 20240402, 20240403, 20240404, 20240405, 20240406, 20240407, 20240408, 20240409, 20240410, 20240411, 20240412, 20240413, 20240414, 20240415, 20240416, 20240417, 20240418, 20240419, 20240420, 20240421, 20240422, 20240423, 20240424, 20240425, 20240426, 20240427, 20240428, 20240429, 20240430, 20240501, 20240502, 20240503, 20240504, 20240505, 20240506, 20240507, 20240508, 20240509, 20240510, 20240511, 20240512, 20240513, 20240514, 20240515, 20240516, 20240517, 20240518, 20240519, 20240520, 20240521, 20240522, 20240523, 20240524, 20240525, 20240526, 20240527, 20240528, 20240529, 20240530, 20240531, 20240601, 20240602, 20240603, 20240604, 20240605, 20240606, 20240607, 20240608, 20240609, 20240610, 20240611, 20240612, 20240613, 20240614, 20240615, 20240616, 20240617, 20240618, 20240619, 20240620, 20240621, 20240622, 20240623, 20240624, 20240625, 20240626, 20240627, 20240628, 20240629, 20240630, 20240701, 20240702, 20240703, 20240704, 20240705, 20240706, 20240707, 20240708, 20240709, 20240710, 20240711, 20240712, 20240713, 20240714, 20240715, 20240716, 20240717, 20240718, 20240719, 20240720, 20240721, 20240722, 20240723, 20240724, 20240725, 20240726, 20240727, 20240728, 20240729, 20240730, 20240731, 20240801, 20240802, 20240803, 20240804, 20240805, 20240806, 20240807, 20240808, 20240809, 20240810, 20240811, 20240812, 20240813, 20240814, 20240815, 20240816, 20240817, 20240818, 20240819, 20240820, 20240821, 20240822, 20240823, 20240824, 20240825, 20240826, 20240827, 20240828, 20240829, 20240830, 20240831, 20240901, 20240902, 20240903, 20240904, 20240905, 20240906, 20240907, 20240908, 20240909, 20240910, 20240911, 20240912, 20240913, 20240914, 20240915, 20240916, 20240917, 20240918, 20240919, 20240920, 20240921, 20240922, 20240923, 20240924, 20240925, 20240926, 20240927, 20240928, 20240929, 20240930, 20241001, 20241002, 20241003, 20241004, 20241005, 20241006, 20241007, 20241008, 20241009, 20241010, 20241011, 20241012, 20241013, 20241014, 20241015, 20241016, 20241017, 20241018, 20241019, 20241020, 20241021, 20241022, 20241023, 20241024, 20241025, 20241026, 20241027, 20241028, 20241029, 20241030, 20241031, 20241101, 20241102, 20241103, 20241104, 20241105, 20241106, 20241107, 20241108, 20241109, 20241110, 20241111, 20241112, 20241113, 20241114, 20241115, 20241116, 20241117, 20241118, 20241119, 20241120, 20241121, 20241122, 20241123, 20241124, 20241125, 20241126, 20241127, 20241128, 20241129, 20241130, 20241201, 20241202, 20241203, 20241204, 20241205, 20241206, 20241207, 20241208, 20241209, 20241210, 20241211, 20241212, 20241213, 20241214, 20241215, 20241216, 20241217, 20241218, 20241219, 20241220, 20241221, 20241222, 20241223, 20241224, 20241225, 20241226, 20241227, 20241228, 20241229, 20241230, 20241231

How many of these dates are prime? 21 in fact as opposed to last year's 18. Here are the primes:

20240107, 20240219, 20240323, 20240327, 20240411, 20240419, 20240531, 20240603, 20240611, 20240723, 20240729, 20240807, 20240819, 20240821, 20240903, 20241017, 20241029, 20241119, 20241121, 20241211, 20241229

With the above list of 2024's 366 days we can investigate primeness and many other number properties. As with 2023 however, there can be no palindromes. This will be the case until 2030 when 20300302 or 2nd March 2030 will break the drought. As for today's number, there is no listing for it in the OEIS but Numbers Aplenty provides some information. It has the following factorisation:$$20240731=7 \times 1327 \times 2179$$Additionally it is a:

• sphenic number since it has three distinct prime factors
• cyclic number since it has no factors in common with its totient (17328168.)
• Duffinian number since it has no factors in common with its sum of divisors (23160320)
• junction number because it is equal to n + sod(n) for n = 20240699 and 20240708.

Using my multipurpose algorithm, running a Jupyter notebook on my laptop (it times out on SageMathCell), I found the following information.

The Collatz Trajectory for 20240731 is:

[20240731, 60722194, 30361097, 91083292, 45541646, 22770823, 68312470, 34156235, 102468706, 51234353, 153703060, 76851530, 38425765, 115277296, 57638648, 28819324, 14409662, 7204831, 21614494, 10807247, 32421742, 16210871, 48632614, 24316307, 72948922, 36474461, 109423384, 54711692, 27355846, 13677923, 41033770, 20516885, 61550656, 30775328, 15387664, 7693832, 3846916, 1923458, 961729, 2885188, 1442594, 721297, 2163892, 1081946, 540973, 1622920, 811460, 405730, 202865, 608596, 304298, 152149, 456448, 228224, 114112, 57056, 28528, 14264, 7132, 3566, 1783, 5350, 2675, 8026, 4013, 12040, 6020, 3010, 1505, 4516, 2258, 1129, 3388, 1694, 847, 2542, 1271, 3814, 1907, 5722, 2861, 8584, 4292, 2146, 1073, 3220, 1610, 805, 2416, 1208, 604, 302, 151, 454, 227, 682, 341, 1024, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1]

There are 107 steps required to reach 1 (see Figure 1).

Figure 1: a logarithmic scale has been used for the vertical axis

The Aliquot Sequence for 20240731 is:

[20240731, 2919589, 1, 0]

The Anti-Divisors of 20240731 are:

[2, 3, 11, 14, 33, 1613, 2654, 4358, 18578, 25097, 30506, 1226711, 3680133, 5783066, 13493821]

The Arithmetic Derivative of 20240731 is 2916075

The Maximum - Minimum Recursive Algorithm for 20240731 produces:

[20240731, 74199753, 86317632, 75326643, 53308665, 83299662, 77499423, 76326633, 53326665, 43299666, 76199733, 86408532, 86308632, 86326632, 64326654, 43208766, 85317642, 75308643, 84308652, 86308632]

The number of steps required is to reach home prime is 6 :

[20240731, 713272179, 31719736091, 180117612291, 3187931952743, 31310185086111, 349092126039593]

The multiplicative persistence of 20240731 is as follows:

[20240731, 0]

20240731 has Odds and Evens Trajectory of length 2 and is:

[20240731, 20240734, 20240732, 20240732]

The multipurpose algorithm that I developed has thus proven very useful for large numbers like 20240731 where the OEIS often provides no information. I should develop it further and set it up permanently on my laptop. I'd like to add results of adding and subtracting sums of digits and products of digits for starters but I need to systematically go through my SageMath notebook and include whatever I think might be interesting. For example, the determinant of the circulant matrix of a number comes to mind. This is an enterprise that I should seriously set about undertaking. It will refamiliarise me with many topics that I haven't had contact with for a while.

Hidden Connections

In a separate blog of mine, I was writing about the introduction of a Central Bank Digital Currency or CBDC as it's most conveniently referred to. Whenever I see letters, I also see numbers. Such is my affliction. The numbers I see lie behind the letters when they are viewed as placeholders for numbers in non-decimal bases. In the case of CBDC, the obvious number base to consider is hexadecimal, base 16.

Bases of 14 and 15 are possible but they are not commonly used whereas hexadecimal is ubiquitous. So what decimal number does a hexadecimal CBDC convert to then. The answer is 52188. What are the hidden connections I refer to? Well, the connection lies in its factorisation which is \( 2^2 \times 3 \times 4349 \).

The connection is only apparent to me because I'm born on April 3rd 1949 which can be written as 4-3-49 in date notation and compressed to 4349. The smaller initial factors \(2^2=4\) and \(3\) reinforce this. I'm not suggesting any deep connection between myself and CBDC.  I'm just pointing this out as a mathematical curiosity. 

It's interesting to me to be able to see numbers lurking behind words and letters. See my recent post: Elements, Letters and Numbers. I also find it interesting to see dates as numbers and numbers as dates. See my recent posts: The Julian Day Number, Special Numbers, Special Dates, Turning Dates Into Numbers.

Getting back to my birthdate and birthdates in general, it can be observed that birthdates or any dates can be interpreted as numbers in the following ways using my birthdate as the example:

• DD - MM - YY with leading zeros and intermediate zeros optionally omitted.
  
  For example, 03 - 04 - 49 --> 30449 or, as it's said, "third of the fourth forty nine" (3449)
  
  
• MM - DD - YY with leading zeros omitted and intermediate zeros optionally omitted.
  
  For example, 04 - 03 - 49 --> 40349 or, as it's said, "fourth of the third forty nine" (4349)
  
  
• DD - MM - YYYY
  
  For example, 03041949 --> 3041949 or, as it's said, "third of the fourth nineteen forty nine" (341949)
  
  
• MM - DD - YYYY
  
  For example, 04 - 03 - 1949 --> 4031949 or, as it's said, "fourth of the third nineteen forty nine" (431949)
  
  
• YYYYMMDD
  
  For example, 19490403 and intermediate zeros are not usually dropped here. It's this format that I focused on in my blog titled Turning Dates into Numbers back in January of 2023.

Of these various representations, only three are prime:

3 - 4 - 49 = 3449 = 3449

3- 04 - 49 = 30449

4 - 3 - 49 = 4349

4 - 03 - 49 = 157 * 257

3 - 4 - 1949 = 3 * 113983

3 - 04 - 1949 = 3 * 43 * 23581

4 - 3 - 1949 = 3 * 7 * 67 * 307

4 - 03 - 1949 = 3 * 1343983

1949 - 04 - 03 = 3 * 167 * 38903