What's Special About Palindrome 27472?

Palindromic numbers occur every century during a millenium and the millenium I'm focused on stretches from 27000 to 27999. Because the first two digits of numbers in this millenium add to 9, the palindromes that arise have the peculiarity that the middle digit is always the arithmetic digital root, with the exception of 27072. Thus we have:

27172 with digital root of 1

27272 with digital root of 2

27372 with digital root of 3

27472 with digital root of 4

27572 with digital root of 5

27672 with digital root of 6

27772 with digital root of 7 

27872 with digital root of 8 

27972 with digital root of 9

I'll soon be 27472 days old and I've gotten into the habit of creating a post for each palindromic day. So what other special properties does this palindrome have? 

• It is a palindrome in base 9 as well $$27472_{10} \rightarrow 41614_{ \, 9}$$This qualifies it for membership in OEIS A180454: numbers that are 5-digit palindromes in at least two bases.
  
  
• It is a d-powerful number, because it can be written as $$27472=2^3 + 7^4 + 4^3 + 7^5 + 2^{13} $$
• It can be written as a sum of two squares in two different ways because it is a product of a power of 2 and two 4k+1 primes:$$ \begin{align} 27472 &= 2^4 \times 17 \times 101\\ &=24^2+ 164^2\\ &=56^2+ 156^2 \end{align}$$
•  27472 is an untouchable number, because it is not equal to the sum of proper divisors of any number.
  
  
• 27472 is a palindrome with exactly six prime factors (counted with multiplicity) and this qualifies it for membership in OEIS A046332 whose members, up to 40000, are:
  
  
  2772, 2992, 6776, 8008, 21112, 21712, 21912, 23632, 23832, 25452, 25752, 25952, 27472, 28782, 29392
  
  
• 27472 is a palindrome which is even and in which the parity of digits alternates. This qualifies it for membership in OEIS A030149. The initial members are:
  
  
  0, 2, 4, 6, 8, 212, 232, 252, 272, 292, 414, 434, 454, 474, 494, 616, 636, 656, 676, 696, 818, 838, 858, 878, 898, 21012, 21212, 21412, 21612, 21812, 23032, 23232, 23432, 23632, 23832, 25052, 25252, 25452, 25652, 25852, 27072, 27272, 27472
  
  
• The Collatz Trajectory for 27472 is:

27472, 13736, 6868, 3434, 1717, 5152, 2576, 1288, 644, 322, 161, 484, 242, 121, 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 108 steps required to reach 1. Figure 1 shows a plot of the numbers using a logarithmic scale for the vertical axis. 

 

Figure 1

• The Aliquot Sequence for 27472 is:

27472, 29444, 25240, 31640, 50440, 73040, 114448, 117680, 156112, 174224, 163366, 121862, 81418, 40712, 46648, 61352, 53698, 26852, 28210, 36302, 25954, 15086, 8794, 4400, 7132, 5356, 4836, 7708, 6404, 4810, 4766, 2386, 1196, 1156, 993, 335, 73, 1, 0

            Figure 2 shows a plot of these numbers. 

Figure 2

• The Anti-Divisors of 27472 are:

3, 5, 7, 9, 11, 15, 27, 32, 33, 37, 45, 47, 55, 99, 111, 135, 165, 167, 185, 297, 329, 333, 407, 495, 544, 555, 999, 1169, 1221, 1485, 1665, 2035, 3232, 3663, 4995, 6105, 7849, 10989, 18315 

• The Arithmetic Derivative of 27472 is 56832
  

• The Maximum - Minimum Recursive Algorithm for 27472 produces:

27472, 54945, 50985, 92961, 86922, 75933, 63954, 61974, 82962, 75933 

• The Minimal Goldbach decomposition of 27472 is 23 and 27449
  

• The number of steps required is to reach home prime is 5 :
• 27472
• 222217101
• 333310925169
• 3365956198099
• 1111271910230901
• 3419034730977487
  

• The multiplicative persistence of 27472 is as follows: 27472, 784, 224, 16, 6
  
  
• 27472 is a Zumkeller number, because its divisors can be partitioned in two sets with the same sum (28458). There are many such partitions, one example of which is:
  
  [8, 34, 136, 808, 27472] and [1, 2, 4, 16, 17, 68, 101, 202, 272, 404, 1616, 1717, 3434, 6868, 13736] both of which sum to 28458
  
  The divisors of 27472 are 1, 2, 4, 8, 16, 17, 34, 68, 101, 136, 202, 272, 404, 808, 1616, 1717, 3434, 6868, 13736 and 27472.
  

• 27472 has Odds and Evens Trajectory of length 1 and is 27472, 27478, 27478
  
  
• 27472 is a pseudoperfect number because it is the sum of a subset of its proper divisors. There are many such subsets one of which is [101, 1616, 1717, 3434, 6868, 13736]. Permalink.
  
  
• 27472 is a practical number, because each smaller number is the sum of distinct divisors of 27472.  For example, take an arbitrary number like 23891. It can expressed as the sum of distinct divisors of 24742 in several different ways e.g. the sum of 1, 17, 68, 272, 404, 808, 1717, 6868 and 13736.
  
  
• 27472 is of course an abundant number, since it is smaller than the sum of its proper divisors (29444).

27472 has the property, shared by all the numbers in its decade, that its digit sum is given by the concatenation of its first and last digit, here 22. Thus:

• 27470 has digit sum 20 
• 27471 has digit sum 21 
• 27472 has digit sum 22
• 27473 has digit sum 23 
• 27474 has digit sum 24 
• 27475 has digit sum 25 
• 27476 has digit sum 26 
• 27477 has digit sum 27 
• 27478 has digit sum 28 
• 27479 has digit sum 29

This permalink will generate a list of all 1230 numbers in the range up to 40000. However, there are only 13 palindromes with this property and they are 191, 2992, 10901, 11711, 12521, 13331, 14141, 25852, 26662, 27472, 28282, 29092 and 39993.

Under Conway's Game of Life rules, the 27472 shape shown at the beginning of this blog stabilises after about 914 generations to the shapes shown in Figures 3 and 4 with the paths of the five gliders visible in Figure 4.

Figure 3

Figure 4