26362: Another Special Palindrome

The day that I turned 26262 days old I dedicated a post to the number and titled it 26262: A Special Palindrome. That was on February 21st 2021. Today I'm enjoying the next successive palindromic day, having turned 26362 days old. 

Now this is a somewhat unusual palindrome in that it is not a member of OEIS A067030. 

A067030

Numbers \(n\) that are of the form \(k\) + reverse(\(k\)) for at least one \(k\).

Let's be clear firstly that such numbers are not common. The 1000th such number is 38772 which translates to a percentage density of less than 2.6%. The first such numbers are:

0, 2, 4, 6, 8, 10, 11, 12, 14, 16, 18, 22, 33, 44, 55, 66, 77, 88, 99, 101, 110, 121, 132, 141, 143, 154, 161, 165, 176, 181, 187, 198, 201, 202, 221, 222, 241, 242, 261, 262, 281, 282, 302, 303, 322, 323, 342, 343, 362, 363, 382, 383, 403, 404, 423, 424, 443 

Of these first 56 members of the sequence, 29 (more than 50%) are palindromic. This is not surprising because most numbers, under the repeated Reverse and Add operation, lead to a palindrome. 196 is the first number that apparently does not. Let's look at the palindromes not belonging to this sequence: 

131, 151, 171, 191, 212, 232, 252, 272, 292, 313, 333, 353, 373, 393, 434

All of them lead to palindromes under Reverse and Add, as shown below:

131 requires 1 steps to reach the palindrome 262

151 requires 2 steps to reach the palindrome 505

171 requires 2 steps to reach the palindrome 585

191 requires 4 steps to reach the palindrome 2552

212 requires 1 steps to reach the palindrome 424

232 requires 1 steps to reach the palindrome 464

252 requires 2 steps to reach the palindrome 909

272 requires 2 steps to reach the palindrome 989

292 requires 8 steps to reach the palindrome 233332

313 requires 1 steps to reach the palindrome 626

333 requires 1 steps to reach the palindrome 666

353 requires 3 steps to reach the palindrome 4444

373 requires 4 steps to reach the palindrome 9559

393 requires 4 steps to reach the palindrome 9339

434 requires 1 steps to reach the palindrome 868

26362 as it turns out is a member of OEIS A070001:

A070001

Palindromes whose 'Reverse and Add' trajectory (presumably) does not lead to another palindrome.

 The initial members of this sequence are:

4994, 8778, 9999, 11811, 19591, 22822, 23532, 23632, 23932, 24542, 24742, 24842, 24942, 26362, 27372, 29792, 29892, 33933, 34543, 34743, 34943, 39493, 44744, 46064, 46164, 46364, 46564, 46964, 47274, 47574, 48284, 48584, 48684, 48884

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:

19591, 23532, 23932, 24542, 24742, 24942, 26362, 27372, 29792, 33933, 34543, 34743, 34943, 39493, 44744, 46164, 46364, 46564, 46964, 47574, 48584

So 26362 is only the 7th palindrome to have the simultaneous property that:

• it cannot be derived from \(k\) + reverse(\(k\)) for one or more values of \(k\)
• its Reverse and Add trajectory (presumably) does not lead to another palindrome

There already two sevens associated with the number because \(7^2\) is a factor, so we have a lucky triple 7.

This is not the first time that I've written about palindromic numbers. Apart from my 26262: A Special Palindrome post, I've written about:

• Palindromic Cyclops Numbers on August 10th 2020
• L-th Order Palindromes on January 24th 2019
• Lychrel Numbers on September 14th 2016
• 22, Reverse and Add on January 7th 2016
• More on AntiDivisors on February 28th 2021
  (mention made of palindromes whose sum of anti-divisors is palindromic).
• Remembering Reverse and Add, Palindromes and Trajectories on June 22nd 2016 

In researching this post, I came across a category of primes known as Palindromic Wing Primes or PMPs defined as "numbers that are primes, palindromic in base 10, and consisting of one central digit surrounded by two wings having an equal amount of identical digits and different from the central one". Examples are:

101

99999199999

333333313333333

7777777777772777777777777

11111111111111111111111111111111411111111111111111111111111111111

Some of these primes are regarded as potential Lychrel candidates and are listed in OEIS A320516 (with the rule that palindromes are ineligible being relaxed):

A320516

Palindromic wing primes that are also Lychrel candidates.

Initial members are:

7774777, 777767777, 77777677777, 99999199999, 1111118111111, 7777774777777, 111111181111111, 333333373333333, 77777777677777777, 99999999299999999, 9999999992999999999, 33333333333733333333333, 77777777777677777777777, 333333333333373333333333333

More information about PWPs can be found here. Palindromes and any numbers can be tested for Lychrel candidature using SageMathCell. A permalink is attached to the screenshoot in Figure 1.

Figure 1: permalink