Another Mid-Millenium Number

On Saturday 23rd of October 2021, I made a post titled Counting People with Mid-Millennium Numbers in which I examined my diurnal age of 26500 on that date. Now, almost three years later, I've reached another "milestone": 27500.

As I said in that post, mid-millennial numbers are popular rounding numbers for populations of towns and islands or communities with common interests or characteristics. These are to be preferred to millennium numbers such as 27,000 or 28000 that appear a little too "approximate" (having only two significant figures as opposed to the three of 27,500). Such numbers are also popular with a wide variety of quantities such as tonnage, money and distances.

The 30 divisors of 27500 are 1, 2, 4, 5, 10, 11, 20, 22, 25, 44, 50, 55, 100, 110, 125, 220, 250, 275, 500, 550, 625, 1100, 1250, 1375, 2500, 2750, 5500, 6875, 13750 and 27500. It's special in the sense that all the digits (apart from 0) are prime. This won't occur again until 30500. As with the previous mid-millennial number, the number appears frequently in population statistics. Here are some examples:

• Published today, the NHS Workforce Race Equality Standard shows Black and minority ethnic (BME) staff make up almost a quarter of the workforce overall (24.2% or 383,706 staff) – an increase of 27,500 people since 2021 (22.4% of staff).
  
  
• The Portuguese Grand Prix of Formula 1, which will be played between Friday and Sunday, will have a maximum capacity of 27,500 spectators, according to the government dispatch published today, October 21, in Diário da República.
  
  
• More than 27,500 people in Gaza have already been killed over the past four months, according to Gaza’s Ministry of Health. Further fighting in Rafah risks claiming the lives of even more people. It also risks further hampering a humanitarian operation already limited by insecurity, damaged infrastructure and access restrictions.
  
  
• The anonymous online study, ‘The Global Brain Health Survey’, involved more than 27,500 people worldwide and was led by the Norwegian Institute of Public Health in collaboration with the University of Oslo. 
  
  
• Kirchberg’s resident population is expected to grow almost six-fold over the next 20 years, according to projections from the Fonds Kirchberg.The body responsible for coordinating the development of one of the capital’s business districts forecasts that there will be 23,700 people living in the area by 2040, up from 4,000 in 2020. It suggests that the district’s maximum capacity would be capped at 27,500 beyond 2040.
  
  
• The Kenyan government says it has set up more than 100 camps to house over 27,500 people displaced by flooding.According to government data, more than 190,000 people have so far been affected by the floods and at least 210 are known to have died.
  

The aliquot sequence for 27500 has a length of 206 steps which are:

[27500, 38104, 40016, 40708, 30538, 15272, 14968, 13112, 13888, 18624, 31160, 44440, 65720, 89800, 119450, 102820, 119444, 105760, 144476, 121804, 97380, 198552, 297888, 518592, 909904, 998456, 889384, 795416, 774784, 768986, 444454, 261146, 141274, 100934, 52186, 27194, 13600, 21554, 13306, 6656, 7666, 3836, 3892, 3948, 6804, 13580, 19348, 19404, 42840, 125640, 283860, 633420, 1562004, 2535180, 5206260, 9371436, 12495276, 20190804, 26921100, 55087540, 60803732, 56587948, 45117684, 69280236, 116780184, 208518216, 312777384, 469166136, 772745304, 1187955816, 1781933784, 2716157736, 4851795384, 8337024936, 14614443864, 27722614536, 48023931924, 82013245164, 134692975476, 205780934846, 107492498242, 53746249124, 42348933724, 35852825156, 27118586620, 29830445324, 24116652916, 18087489694, 10639699874, 5379726934, 2706299954, 1471210894, 735605450, 632620780, 816656420, 898322104, 809756216, 781610824, 685415096, 602212744, 711287846, 355643926, 246590714, 123608986, 61804496, 72770224, 68943920, 91350880, 128146160, 170324320, 242956688, 264953512, 248965388, 248965444, 290225852, 310243108, 343261436, 345226084, 363478556, 363478612, 383892908, 438979156, 520540076, 520540132, 539131250, 616169806, 498886994, 249443500, 314159540, 346712980, 406492340, 486797260, 537866996, 403400254, 201700130, 166294750, 145009490, 131361862, 68478170, 54782554, 27444794, 17643046, 8821526, 6384874, 3696566, 1888594, 944300, 1555540, 2282924, 2282980, 3442460, 4965604, 5062876, 6042092, 6693988, 8128904, 9877396, 8355308, 7779412, 5834566, 2942234, 1471120, 2600048, 3337072, 3287504, 3661456, 3432646, 2557142, 1826554, 1027814, 519394, 259700, 408226, 345758, 246994, 164846, 111634, 55820, 61444, 46090, 44630, 35722, 19034, 10534, 6026, 3478, 1994, 1000, 1340, 1516, 1144, 1376, 1396, 1054, 674, 340, 416, 466, 236, 184, 176, 196, 203, 37, 1, 0]

With a logarithmic vertical axis, the trajectory has a pleasing mountain-like appearance. See Figure 1.

Figure 1

The Collatz trajectory of 27500 has 90 steps and, with a logarithmic vertical axis, its trajectory has a noticeably jagged appearance (see Figure 2):

[27500, 13750, 6875, 20626, 10313, 30940, 15470, 7735, 23206, 11603, 34810, 17405, 52216, 26108, 13054, 6527, 19582, 9791, 29374, 14687, 44062, 22031, 66094, 33047, 99142, 49571, 148714, 74357, 223072, 111536, 55768, 27884, 13942, 6971, 20914, 10457, 31372, 15686, 7843, 23530, 11765, 35296, 17648, 8824, 4412, 2206, 1103, 3310, 1655, 4966, 2483, 7450, 3725, 11176, 5588, 2794, 1397, 4192, 2096, 1048, 524, 262, 131, 394, 197, 592, 296, 148, 74, 37, 112, 56, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1]

Figure 2

Here are some other interesting facts about the number:

• The Anti-Divisors of 27500 are [3, 7, 8, 9, 21, 27, 40, 63, 81, 88, 97, 189, 200, 291, 440, 567, 679, 873, 1000, 2037, 2200, 2619, 5000, 6111, 7857, 11000, 18333]
  
  
• The Arithmetic Derivative of 27500 is 52000
  
  
• The Maximum - Minimum Recursive Algorithm for 27500 produces [27500, 74943, 62964, 71973, 83952, 74943]
  
  
• The Minimal Goldbach decomposition of 27500 is 13 and 27487
  
  
• The number of steps required is to reach home prime is 7:
  [27500, 22555511, 1110511951, 3313355021, 31337105733, 3373163729137, 4936768328311, 101312973757451]
  
  
• 27500 has Odds and Evens Trajectory of length 12 and is [27500, 27510, 27521, 27530, 27543, 27552, 27565, 27574, 27587, 27596, 27609, 27617, 27624, 27617]

For the previous mid-millennial day see Counting People with Mid-Millennium Numbers.

26662

Well, not only is my diurnal age today (April 2nd 2022) an impressive palindrome (26662) but the date also marks my 73rd solar return. This is the day that the Sun returns to the exact position that it occupied at the time of my birth, namely 12°47'07" of Aries. The date of the solar return is always close to a person's official birthday but not necessarily the same as it. In my case, April 3rd is my official birthday.

I'll use this number as an excuse to revisit some mathematical topics that I haven't visited in quite a while. Let's begin:

COLLATZ TRAJECORY

What is the trajectory of 26662 under the 3\(x\)+1 recursive algorithm? The algorithm is applied to any positive integer \(n\):

\(n\) → \(n\)/2 (n is even)

\(n\) → 3\(n\) + 1 (n is odd)

Most, but not all, numbers reach 1. 26662 is no exception but it takes 95 steps. It's trajectory is:

26662, 13331, 39994, 19997, 59992, 29996, 14998, 7499, 22498, 11249, 33748, 16874, 8437, 25312, 12656, 6328, 3164, 1582, 791, 2374, 1187, 3562, 1781, 5344, 2672, 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

Figure 1 shows a plot of these values:

Figure 1

ALIQUOT SEQUENCE

The aliquot sequence for an integer \(n\) is obtained from the recurrence relation:$$a_n=\sigma(a_{n-1})-a_{n-1}$$where \( \sigma(n) \) is the sum of divisors function. Most aliquot number sequences reach 0 but if an amicable number is encountered then it ends up alternating between the abundant and deficient member of the amicable pair. Such is the case with 26662 where the amicable number pair (2924, 2620) is reached. Here is the trajectory:

26662 --> 13334 --> 7186 --> 3596 --> 3124  --> 2924 --> 2620 --> 2924

ANTI-DIVISORS

While 26662 may only have four divisors (1, 2, 13331 and 26662), it has a larger number of anti-divisors. These are:

3, 4, 5, 9, 15, 25, 27, 45, 75, 79, 135, 225, 237, 395, 675, 711, 1185, 1975, 2133, 3555, 5925, 10665, 17775

I've made two posts about anti-divisors. One is titled Anti-divisors on February 26th 2016 and another, more comprehensive, post on February 28th 2021 titled More on Anti-divisors.

ARITHMETIC DERIVATIVE

The arithmetic derivative of a natural number \(n\) is given by:$$ \begin{align} p'&=1 \text{ for any prime }p\\(pq)'&=p'q+pq' \text{ for any } p,q \in \mathbb{N } \end{align}$$The arithmetic derivative of 26662 is thus:$$ \begin{align}  26662'  &= (2 \times 13331)'\\&= 2' \times 13331 + 2 \times 13331'  \\&= 13331+2\\&=13333\end{align}$$

DIGIT MANIPULATION

Suppose we have a number such as 26662 and we want to rearrange its digits in such a way that the maximum and minimum possible numbers are created. This would give us 66622 as a maximum and 22666 as a minimum. Furthermore, let’s suppose we want to subtract the minimum from the maximum to get 43956 and then repeat this digital manipulation repeatedly until some resolution is reached. 

In this case, the trajectory of 26662 looks like this: 

26662 --> 43956 --> 61974 --> 82962 --> 75933 -->  63954 --> 61974

As can be seen, a loop has been entered {61974, 82962, 75933, 63954}. Most numbers enter a loop but sometimes 0 is reached if a number like 999 is encountered.

GOLDBACH DECOMPOSITION

Goldbach’s conjecture states that every even number can be expressed as a sum of two primes. There are many such decompositions for any given number (see my post Goldbach’s Conjecture Revisited) but the one containing the smallest and largest prime is known as the minimal decomposition.

There are 439 Goldback decompositions of 26662

The minimal decomposition is (24593, 2069)

HOME PRIME

The home prime of a number n is the prime reached by concatenating its prime factors (in the order smallest to largest) and repeating until a prime is reached. In the case of 26662, only three steps are required:

26662 = 2 x 13331 --> 213331 = 383 x 557 --> 383557

MULTIPLICATIVE PERSISTENCE

The multiplicative persistence of a number counts the number of steps required to reach a fixed number (often zero) when the digits are multiplied together. 26662 has a multiplicative persistence of 4:

26662 --> 864 --> 192 --> 18 --> 8

A variation of this, that I came up with, is to add this product to the original number. So the 864 from the initial product of digits is added to 26662 to give 27526 and this process is repeated until a number with a zero is reached, after which there can be no further change. The trajectory for 26662 is:

26662 --> 27526 --> 28366 --> 30094

ODDS 'n EVENS

This algorithm takes a number and applies the following recursive process to the number: add the sum of its odd digits to the number and subtract the sum of the even digits, repeating this process until a stable number is reached.  The trajectory of 26662 is as follows:

26662 --> 26640 --> 26622 --> 26604 --> 26586 --> 26569 --> 26569

Thus 26662 is a captive of the attractor 26569 that has a total of 92 captives.

PALINDROME

26662 is of course a palindrome but it has the interesting property that its two prime factors (2 and 13331) are also palindromic (2 of course trivially so). 26662 is also palindromic in base 8 (64046).

I'll leave off there but this post was useful for me in that I was reminded of many concepts that I hadn't encountered in quite a while. I simply made my way through my Google document where I've recorded these concepts over the past few years and applied them to this particular number. It's something that I should do on a regular basis, perhaps even writing a program that would automatically generate this information for a given number. That would be useful!

ADDENDUM: added April 7th 2022

In fact, I did write such a program as alluded to in the paragraph above. Here is a permalink to SageMathCell using 26667 as an example. The output is:

The Collatz Trajectory for 26667 is:

[26667, 80002, 40001, 120004, 60002, 30001, 90004, 45002, 22501, 67504, 33752, 16876, 8438, 4219, 12658, 6329, 18988, 9494, 4747, 14242, 7121, 21364, 10682, 5341, 16024, 8012, 4006, 2003, 6010, 3005, 9016, 4508, 2254, 1127, 3382, 1691, 5074, 2537, 7612, 3806, 1903, 5710, 2855, 8566, 4283, 12850, 6425, 19276, 9638, 4819, 14458, 7229, 21688, 10844, 5422, 2711, 8134, 4067, 12202, 6101, 18304, 9152, 4576, 2288, 1144, 572, 286, 143, 430, 215, 646, 323, 970, 485, 1456, 728, 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 170 steps required to reach 1

The Aliquot Sequence for 26667 is:

[26667, 11865, 10023, 4425, 3015, 2289, 1231, 1, 0]

The Anti-Divisors of 26667 are:

[2, 5, 6, 7, 18, 19, 133, 401, 2807, 5926, 7619, 10667, 17778]

The Arithmetic Derivative of 26667 is 17787

The Maximum - Minimum Recursive Algorithm for 26667 produces:

[26667, 49995, 53955, 59994, 53955]

There are no Goldback Decompositions of 26667 because it is odd.

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

[26667, 332963, 378999, 33334679, 733114387, 2969246923]

The multiplicative persistence of 26667 is as follows:

[26667, 3024, 0]

26667 has Odds and Evens Trajectory of length 6 and is:

[26667, 26654, 26641, 26624, 26604, 26586, 26569, 26569]

Free Fibonacci Sequences

On turning 26404 days, I couldn't help but notice the 404, a number made famous by the experience of everyone who has searched the Internet and failed to find what was being sought.

Apart from containing 404 as a subset of its digits, 26404 has some other interesting properties. Foremost amongst these is the fact that it is a member of OEIS A008892.

A008892

Aliquot sequence starting at 276.

I wrote about aliquot sequences in an eponymous post of December 20th 2017 and again, only recently, in Aliquot Sequences Revisited on June 21st 2021. 276 is the first of a sequence of numbers that are not known to be finite or periodic when the aliquot algorithm is applied. This algorithm takes as its input any integer \(n\) and returns the sum of the number's aliquot parts or proper divisors, \( \sigma(n)-n\). This output serves as the new input and the process is repeated until, most commonly, a prime number \(p\) is reached. Since \( \sigma(p)-p=1\), this means the process terminates because \( \sigma(1)=0\).

For some numbers, as far as can be determined, the process never terminates. These numbers include:

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... and forming OEIS A131884 

In the case of 276, the sequence of numbers up to and including 26404 is:

276, 396, 696, 1104, 1872, 3770, 3790, 3050, 2716, 2772, 5964, 10164, 19628, 19684, 22876, 26404

However, this post is mainly about another interesting property of 26404 and that is its membership of OEIS A232666:

A232666

6-free Fibonacci numbers.

The OEIS comment is that:

The sequences of \(n\)-free Fibonacci numbers were suggested by John H. Conway. \(a(n)\) is the sum of the two previous terms divided by the largest possible power of 6. The sequence coincides with the Fibonacci sequence until the first multiple of 6 in the Fibonacci sequence: 144, which in this sequence is divided by 36 to produce 4.

The sequence of numbers leading to 26404 in OEIS A232666 is:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 4, 93, 97, 190, 287, 477, 764, 1241, 2005, 541, 2546, 3087, 5633, 8720, 14353, 23073, 37426, 60499, 97925, 26404

Here is the permalink to the generation of this sequence. There is a paper titled Free Fibonacci Sequences by Brandon Avila and Tanya Khovanova that appears in the Journal of Integer Sequences (Vol. 17 (2014), Article 14.8.5) that analyses these sequences in some detail. The authors of the paper write:

Let us denote Fibonacci numbers by \(F_k\). We define our indices such that \(F_0 = 0\) and \(F_1 = 1\). The sequence is defined by the Fibonacci recurrence: \(F_{n+1} = F_n + F_{n−1} \) (see OEIS A000045). We call an integer sequence \(a_n\) Fibonacci-like if it satisfies the Fibonacci recurrence: \(a_k = a_{k−1} + a_{k−2}\). A Fibonacci-like sequence is similar to the Fibonacci sequence, except that it starts with any two integers. The second most famous Fibonacci-like sequence is the sequence of Lucas numbers \(L_i\) that starts with \(L_0 = 2\) and \(L_1 = 1\): \(2, 1, 3, 4, 7, 11, \dots \) (see OEIS A000032). 

An \(n\)-free Fibonacci sequence starts with any two integers, \(a_1\) and \(a_2\), and is defined by the recurrence:$$a_k = \frac{a_{k−1} + a_{k−2}}{n^{ν_n(a_{k−1}+a_{k−2})}}$$where \(ν_n(x)\) is the exponent of the largest power of \(n\) that is a divisor of \(x\). To continue the tradition, we call numbers in the \(n\)-free Fibonacci sequence that starts with \(a_0 = 0\) and \(a_1 = 1\) \(n\)-free Fibonacci numbers.

The authors then go on to look at a variety of \(n\)-free Fibonacci sequences. They start with 2-free Fibonacci sequences and find that they all end in a cycle of length 1. For example, starting with \(a_0=5\) and \(a_1=10\) gives:$$5, 10, 15, 25, 5, 15, \overbrace{5}, \dots$$For 3-free Fibonacci sequences, it is suspected that all end in a cycle of \(k, k, 2k\) but it has not been proven. For example, starting with \(a_0=3\) and \(a_1=7\) gives:$$3, 7, 10, 17, 1, 2, \overbrace{1, 1, 2}, \dots$$with 1, 1, 2 repeating. Similarly, taking \(a_0=13\) and \(a_1=7\), the sequence generated is$$13, 7, 20, 1, 7, 8, 5, 13, 2, 5, 7, 4, 11, 5, 16, 7, 23, 10, 11, 7, 2, \overbrace{1, 1, 2} \dots$$ with 1, 1, 2 again repeating. The authors then go on to say that:

Consider the 4-free Fibonacci sequence starting with 0, 1. This sequence is OEIS A224382: 0, 1, 1, 2, 3, 5, 2, 7, 9, 1, 10, 11, 21, 2, 23, 25, .... It seems that this sequence grows and does not cycle. In checking many other 4-free Fibonacci sequences, we still did not find any cycles. The behaviour of 4-free sequences is completely different from the behaviour of 3-free sequences. For 3-free sequences, we expected that all of them cycle. Here, it might be possible that none of them cycles.

Let us look at the Lucas sequence mod 5: 2, 1, 3, 4, 2, 1, ... and see that no term is divisible by 5. Clearly, no term in the Lucas sequence will require that we factor out a power of 5, and the terms will grow indefinitely. Thus, the Lucas sequence is itself a 5-free Fibonacci sequence. On the other hand, it becomes quickly evident that the sequence of 5-free Fibonacci numbers: 0, 1, 1, 2, 3, 1, 4, 1, 1, 2, ... (see OEIS A214684) cycles. Some sequences cycle, and some clearly do not!

John Conway: 1937 - 2020

At the beginning of the paper, it's said that "John Horton Conway likes playing with the Fibonacci sequence. Instead of summing the two previous terms, he sums them up and then adds a twist: some additional operation." Of course, at that time, John Conway was still alive. He only died on April 11th 2020. That's a good way of thinking about the free Fibonacci sequences: Fibonacci with a twist!

I've posted frequently about Fibonacci sequences:

• The Fibonacci Number Base on July 30th 2019
• Fibonacci and Finance on April 15th 2019
• Random Fibonacci Numbers on March 9th 2020
• Dying Rabbits on June 26th 2021
• Fibonacci-like Sequences on June 10th 2020
• Finding Fibonacci and Tribonacci Seed Numbers on July 12th 2019
• Beyond Fibonacci on March 10th 2019
• Vishwanath Number on March 8th 2019
• Goldbach's Conjecture and Zeckendorf's Theorem on November 15th 2015
• Heptagonal Numbers on April 22nd 2019
• Metallic Means on March 3rd 2019

The paper contains more detailed information but the takeaway is that there is plenty of scope for further investigation of Fibonacci-like sequences with a twist. Consider this "twist" on the tribonacci sequence with initial terms of 0, 1 and 2. Each successive term is the sum of the previous three terms but (and here's the twist), if the result is a composite number, replace the term with the composite number's highest prime factor. The first terms are:

0, 1, 2, 3, 3, 2, 2, 7, 11, 5, 23, 13, 41, 11, 13, 13, 37, 7, 19, 7, 11, 37, 11, 59, 107, 59, 5, 19, 83, 107, 19, 19, 29, 67, 23, 17, 107, 7, ...

So what's going on. Well, a plot of the first 400 terms reveals the story. See Figure 1.

Figure 1: permalink

This composite number to highest prime factor "twist" is just something that popped into my head and it instantly yielded a most interesting result: a quick spike and then settling into a cycle after 255 terms. Figure 2 shows the first 1000 terms and the cycles are evident.

Figure 2: permalink

The dramatic rise and fall of the terms and their settling into an endless loop, with its own dramatic peak, are unexpected but that is what happens.

Aliquot Sequences Revisited

I only have one post dealing with Aliquot Sequences and that eponymous post appeared on December 20th 2017 with an update on July 17th 2020. Let's recall that an aliquot sequence is:

A sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.

On December 20th 2017, I turned 25908 days old and this number is a member of OEIS A008888:

A008888

Aliquot sequence starting at 138.

The reason for my update on July 17th 2020 is that on that day I turned 26038 days old and this number is also a member of OEIS A008888 but it appears near the end of the sequence instead of at the beginning like 25908. Here is the full sequence with the two numbers marked in bold:

138, 150, 222, 234, 312, 528, 960, 2088, 3762, 5598, 6570, 10746, 13254, 13830, 19434, 20886, 21606, 25098, 26742, 26754, 40446, 63234, 77406, 110754, 171486, 253458, 295740, 647748, 1077612, 1467588, 1956812, 2109796, 1889486, 953914, 668966, 353578, 176792, 254128, 308832, 502104, 753216, 1240176, 2422288, 2697920, 3727264, 3655076, 2760844, 2100740, 2310856, 2455544, 3212776, 3751064, 3282196, 2723020, 3035684, 2299240, 2988440, 5297320, 8325080, 11222920, 15359480, 19199440, 28875608, 25266172, 19406148, 26552604, 40541052, 54202884, 72270540, 147793668, 228408732, 348957876, 508132204, 404465636, 303708376, 290504024, 312058216, 294959384, 290622016, 286081174, 151737434, 75868720, 108199856, 101437396, 76247552, 76099654, 42387146, 21679318, 12752594, 7278382, 3660794, 1855066, 927536, 932464, 1013592, 1546008, 2425752, 5084088, 8436192, 13709064, 20563656, 33082104, 57142536, 99483384, 245978376, 487384824, 745600776, 1118401224, 1677601896, 2538372504, 4119772776, 8030724504, 14097017496, 21148436904, 40381357656, 60572036544, 100039354704, 179931895322, 94685963278, 51399021218, 28358080762, 18046051430, 17396081338, 8698040672, 8426226964, 6319670230, 5422685354, 3217383766, 1739126474, 996366646, 636221402, 318217798, 195756362, 101900794, 54202694, 49799866, 24930374, 17971642, 11130830, 8904682, 4913018, 3126502, 1574810, 1473382, 736694, 541162, 312470, 249994, 127286, 69898, 34952, 34708, 26038, 13994, 7000, 11720, 14740, 19532, 16588, 18692, 14026, 7016, 6154, 3674, 2374, 1190, 1402, 704, 820, 944, 916, 694, 350, 394, 200, 265, 59, 1, 0

Today is different however, because 26376 (my diurnal age on the date of this post) is not a member of a terminating aliquot sequence like OEIS A008888 but instead it is a member of an aliquot sequence for which it has not yet been determined whether there is an end or an eventual repetition. The sequence is OEIS A014361:

A014361

Aliquot sequence starting at 564.

The initial members of this sequence are 564, 780, 1572, 2124, 3336, 5064, 7656, 13944 and 26376. Numbers like 564 form their own sequence and that is OEIS A131884:

A131884

Numbers conjectured to have an infinite, aperiodic, aliquot sequence.

The initial members of this sequence are:

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, 1512, 1560, 1572, 1578, 1590, 1632, 1650, 1662, 1674, 1722, 1734, 1758, 1770, 1806, 1836

So I thought that today's number with its membership of a rather exclusive club was worth a mention.

Aliquot Sequences

My attention was drawn to aliquot sequences today, day 25098, because the number is a member of four aliquot sequences (as shown below):

From any starting point, it's easy enough to calculate the next member in the sequence by using the divisor function \(\sigma_1 \). For example, the second term in the sequence starting with 138 is \(\sigma_1 (138) -138=150\). Many sequences lead to a prime number and then terminate because \(\sigma_1 (\text{prime number}) -\text{prime number}=1\) and \(\sigma_1(1) -1=0\). The sequence beginning with 138 (OEIS A008888) has 178 members and ends in 59, 1, 0. OEIS A008889 is really the same as OEIS A008888 except for the starting point (150 instead of 138)

Aliquot sequence OEIS A008890 is different however, and starts with 168 but the second term is 312 which is the fifth term in OEIS A008888. Aliquot sequence OEIS A074907 starts with 570 but after a few terms reaches 19434 which again is a term in the OEIS A08888 sequence.

Not all aliquot sequences end. To quote from Wikipedia:

There are a variety of ways in which an aliquot sequence might not terminate:

• A perfect number has a repeating aliquot sequence of period 1. The aliquot sequence of 6, for example, is 6, 6, 6, 6, ... 

• An amicable number has a repeating aliquot sequence of period 2. For instance, the aliquot sequence of 220 is 220, 284, 220, 284, ... 

• A sociable number has a repeating aliquot sequence of period 3 or greater. (Sometimes the term sociable number is used to encompass amicable numbers as well.) For instance, the aliquot sequence of 1264460 is 1264460, 1547860, 1727636, 1305184, 1264460, ... 

• Some numbers have an aliquot sequence which is eventually periodic, but the number itself is not perfect, amicable, or sociable. For instance, the aliquot sequence of 95 is 95, 25, 6, 6, 6, 6, ... . Numbers like 95 that are not perfect, but have an eventually repeating aliquot sequence of period 1 are called aspiring numbers (OEIS  A063769).

Numbers whose Aliquot sequence is not known to be finite or eventually periodic are:

276, 306, 396, 552, 564, 660, 696, 780, 828, 888, 966, 996, 1074, 1086, 1098, 1104, 1134, 1218, 1302, 1314, 1320, 1338, 1350, 1356, 1392, 1398, 1410, 1464, 1476, 1488, ... (sequence A131884 in the OEIS) 

ADDENDUM: 17th July 2020

Today, I turned 26038 days old and I revisited OEIS A008888 (Aliquot sequence starting at 138) because this number is a member of the sequence and appears very near the end. Here is the full sequence:

138, 150, 222, 234, 312, 528, 960, 2088, 3762, 5598, 6570, 10746, 13254, 13830, 19434, 20886, 21606, 25098, 26742, 26754, 40446, 63234, 77406, 110754, 171486, 253458, 295740, 647748, 1077612, 1467588, 1956812, 2109796, 1889486, 953914, 668966, 353578, 176792, 254128, 308832, 502104, 753216, 1240176, 2422288, 2697920, 3727264, 3655076, 2760844, 2100740, 2310856, 2455544, 3212776, 3751064, 3282196, 2723020, 3035684, 2299240, 2988440, 5297320, 8325080, 11222920, 15359480, 19199440, 28875608, 25266172, 19406148, 26552604, 40541052, 54202884, 72270540, 147793668, 228408732, 348957876, 508132204, 404465636, 303708376, 290504024, 312058216, 294959384, 290622016, 286081174, 151737434, 75868720, 108199856, 101437396, 76247552, 76099654, 42387146, 21679318, 12752594, 7278382, 3660794, 1855066, 927536, 932464, 1013592, 1546008, 2425752, 5084088, 8436192, 13709064, 20563656, 33082104, 57142536, 99483384, 245978376, 487384824, 745600776, 1118401224, 1677601896, 2538372504, 4119772776, 8030724504, 14097017496, 21148436904, 40381357656, 60572036544, 100039354704, 179931895322, 94685963278, 51399021218, 28358080762, 18046051430, 17396081338, 8698040672, 8426226964, 6319670230, 5422685354, 3217383766, 1739126474, 996366646, 636221402, 318217798, 195756362, 101900794, 54202694, 49799866, 24930374, 17971642, 11130830, 8904682, 4913018, 3126502, 1574810, 1473382, 736694, 541162, 312470, 249994, 127286, 69898, 34952, 34708, 26038, 13994, 7000, 11720, 14740, 19532, 16588, 18692, 14026, 7016, 6154, 3674, 2374, 1190, 1402, 704, 820, 944, 916, 694, 350, 394, 200, 265, 59, 1, 0.

This is not the last time I'll encounter OEIS A008888 because in a couple of years time, I'll meet 26742 and 26754 (assuming I'm still alive).