26262: A Special Palindrome

Indeed, "There are things that drift away, like our endless numbered days" and 26262 is one of them. Today I turned 26262 days old and this number should pass away perhaps only after it has received its due attention. 

So what makes this particular palindrome special? Well, to begin with, it's sandwiched between two primes and that doesn't happen often as OEIS A113838 reveals.

A113838

Palindromes sandwiched between twin primes.

The members of this sequence, up to and including 26262 are: 

4, 6, 282, 828, 858, 2112, 21012, 21612, 23832, 26262

The next property involves the Reverse and Add operation that I've written about before.

A222533

Smallest palindrome which requires at least \(n\) iterations of Reverse and Add to reach a palindrome.

In the case of 26262, 31 iterations are required. The smallest number (that is not a palindrome) that requires 31 steps to reach a palindrome under the Reverse and Add operation is 13293.

An interesting thing happens if 26262 is concatenated with itself to form the number 2626226262. It turns out this number is sandwiched between twin primes as well.

A235109

Averages q of twin prime pairs, such that q concatenated to q is also the average of a twin prime pair.

Yes, 2626226261 and 2626226263 are twin primes and 2626226262 is right in the middle!

This property had me scratching my head as you may well do when you read it.

A045960

Palindromic even lucky numbers.

Ah, lucky numbers are all odd and thus there are no even lucky numbers. Well, there are IF you start with only the even numbers (2, 4, 6 etc.) and apply a similar sieve process to what is applied to the standard lucky numbers. The sieve process works like this:

Write down even numbers: 2 4 6 8 ...; first term > 2 is 4 so starting from 2 remove every 4th number: 2 4 6 10 12 14 18...; next number is 6 so remove every 6th term starting from 2: 2 4 6 10 12 18 20 22 26 etc. 

This will generate OEIS A045954, the even lucky numbers:

A045954

Even-Lucky-Numbers (ELN): generated by a sieve process like that for Lucky  numbers but starting with even numbers.

The initial members are:

2, 4, 6, 10, 12, 18, 20, 22, 26, 34, 36, 42, 44, 50, 52, 54, 58, 68, 70, 76, 84, 90, 98, 100, 102, 108, 114, 116, 118, 130, 132, 138, 140, 148, 150, 164, 170, 172, 178, 182, 186, 196, 198, 212, 214, 218, 228, 230, 234, 244, 246, 260, 262, 268, 278, 282, 290, 298, 300, 308, ...

If you include ONLY palindromic numbers, then you end up with OEIS A045960:

A045960

Palindromic even lucky numbers.

The initial members of this sequence 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, ...

The final property that we'll look at is the fact that 26262 is an undulating Harshad number.

A129120

Undulating Harshad numbers: numbers divisible by the sum of their own digits with decimal expansions in an abab...ab pattern.

The initial members of this sequence are:

10, 12, 18, 20, 21, 24, 27, 30, 36, 40, 42, 45, 48, 50, 54, 60, 63, 70, 72, 80, 81, 84, 90, 171, 252, 414, 828, 1010, 1212, 1818, 2020, 2424, 3030, 3636, 4040, 4848, 5050, 5454, 6060, 7070, 7272, 8080, 9090, 10101, 13131, 20202, 23232, 26262, 30303, 39393, ...

The members that are palindromic are marked in bold. 

There are many more properties of 26262 but I'll just include these five properties as a way of remembering this particular palindrome. My next will occur with 26362.