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]