27372: Another Palindromic Day

Days like today, when I turn 27372 days old, pop up every one hundred days during the course of a millennium of days and there is a 110 day gap between millennia. So, for example, from 27972 to 28082, there will be a gap of 110 days. Today's number shares some important properties with another palindrome, 26362, that I created a post about on June 6th 2021. It was titled 26362: Another Special Palindrome. 

One property that the two share is that they are both members of OEIS  A070001:

A070001

Palindromic integers > 0, whose 'Reverse and Add!' trajectory (presumably) does not lead to another palindrome.

Up to 40000, the members of this sequence are not numerous and they are:

4994, 8778, 9999, 11811, 19591, 22822, 23532, 23632, 23932, 24542, 24742, 24842, 24942, 26362, 27372, 29792, 29892, 33933, 34543, 34743, 34943, 39493

It can be seen that 26362 and 27372 are consecutive and 1010 days apart in terms of my diurnal age. As I wrote in the post previously alluded to:

These palindromes are not regarded as potential Lychrel numbers because they are already palindromes and some of them are the result or end point of \(k\) + reverse(\(k\)) iterations. However, some are not and these, I think, deserve special consideration. These are:

19591, 23532, 23932, 24542, 24742, 24942, 26362, 27372, 29792, 33933, 34543, 34743, 34943, 39493

So 26362 and 27372 are paired again and they are only the 7th and 8th palindromes to have the simultaneous property that:

• they cannot be derived from \(k\) + reverse(\(k\)) for one or more values of \(k\)
• their Reverse and Add trajectories (presumably) do not lead to another palindrome 

These two numbers are also members of OEIS A045960:

A045960

Palindromic even lucky numbers.

Up to 40000, the initial members are:

2, 4, 6, 22, 44, 212, 262, 282, 434, 474, 646, 666, 818, 838, 868, 2442, 2662, 2772, 4884, 4994, 6666, 6886, 8118, 8338, 20202, 20402, 21012, 21812, 22322, 22422, 22922, 23332, 23532, 24042, 25652, 26162, 26262, 26562, 26762, 27372, 28682

A property that 27372 doesn't share with 26762 is that the former's arithmetic digital root is the same of its middle digit. Of the three and five digit palindromes in the range up to 40000, there are only 36 that satisfy this condition. They are (permalink):

919, 929, 939, 949, 959, 969, 979, 989, 999, 18181, 18281, 18381, 18481, 18581, 18681, 18781, 18881, 18981, 27172, 27272, 27372, 27472, 27572, 27672, 27772, 27872, 27972, 36163, 36263, 36363, 36463, 36563, 36663, 36763, 36863, 36963

For example, the arithmetic digital root of 27372 is 2 + 7 + 3 + 7 + 2 = 21 and 2 + 1 = 3. The middle digit of 27372 is 3.