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.