Palindromic Day 27972

Today is palindromic day 27972 and the last palindromic day of the current millenium. The next palindromic day will occur in the new millenium and will 28082, one hundred and ten days from now.

My first observation is that the digits on either side of the central 9 also add to 9 to give a 9 - 9 - 9 pattern with 9 being the arithmetical digital root as well. This is the first and last time that such a triple pattern will occur in the current millenium. It cannot occur in the next two millenia (2 8 x 8 2 and 2 9 x 9 2) because the digits on either side of the central digit add to 10 and 11 respectively. So already 27972 is rather special. 

27972 is also a member of OEIS A344422: palindromes having more divisors than all smaller palindromes. The table below lists the initial members of this sequence and it can be seen that 27972 has a record 48 divisors.

  number   factorisation        divisors

  1        1                    1

  2        2                    2

  4        2^2                  3

  6        2 * 3                4

  44       2^2 * 11             6

  66       2 * 3 * 11           8

  252      2^2 * 3^2 * 7        18

  2112     2^6 * 3 * 11         28

  2772     2^2 * 3^2 * 7 * 11   36

  6336     2^6 * 3^2 * 11       42

  27972    2^2 * 3^3 * 7 * 37   48

  48384    2^8 * 3^3 * 7        72

27972 is also a member of OEIS A020485: least positive palindromic multiple of \(n\), or 0 if none exists. Here are the multiples for the initial values of \(n\).

  count   n       multiple

  1       1       1

  2       2       1

  3       3       1

  4       4       1

  5       5       1

  6       6       1

  7       7       1

  8       8       1

  9       9       1

  10      0       0

  11      11      1

  12      252     21

  13      494     38

  14      252     18

  15      525     35

  16      272     17

  17      272     16

  18      252     14

  19      171     9

  20      0       0

  21      252     12

  22      22      1

  23      161     7

  24      696     29

  25      525     21

  26      494     19

  27      999     37

  28      252     9

  29      232     8

  30      0       0

  31      434     14

  32      2112    66

  33      33      1

  34      272     8

  35      525     15

  36      252     7

  37      111     3

  38      494     13

  39      585     15

  40      0       0

  41      656     16

  42      252     6

  43      989     23

  44      44      1

  45      585     13

  46      414     9

  47      141     3

  48      2112    44

  49      343     7

  50      0       0

  51      969     19

  52      676     13

  53      212     4

  54      27972   518

  55      55      1

  56      616     11

  57      171     3

  58      232     4

  59      767     13

  60      0       0

  61      26962   442

27972 requires 13 steps to reach the palindrome 4964444694 under the Reverse and Add algorithm. 

27972 is a decagonal or ten-sided number and is the 84th decagonal number and the second non-trivial palindromic decagonal number after 232. See Figure 1 where 232 is shown but not 27972.

Figure 1: source

27972 is a member of OEIS A356854: palindromes that can be written in more than one way as the sum of two distinct palindromic primes. In the case of 27972 we have:

10501 + 17471 = 27972

11311 + 16661 = 27972

11411 + 16561 = 27972

12421 + 15551 = 27972