What's Special About 26962?

I'm following my tradition of making a special post for my palindromic days. The last one was 26862 and titled What's Special About 26862? Now I've reached 26962 and this palindrome has some interesting properties. The first is that it can be split up into two factors, each of which is the reverse of the other (with square numbers such as 121 excluded).

26962 = 122 x 221 = 221 x 122

There are not many palindromes with this property in the range up to one million. Here is the list:

• 252 =12 x 21 = 21 x 12
• 20502 = 102 x 201 = 201 x 102
• 23632 = 112 x 211 = 211 x 112
• 26962 = 122 x 221 = 221 x 122

As can be seen, 26962 is last palindrome with this property in the range up to one million. We have to extend the range in order to find more. In the range between one and two million, the following palindromes are found (permalink):

• 1113111 = 1011 x 1101 = 1101 x 1011
• 1226221 = 1021 x 1201 = 1201 x 1021
• 1357531 = 1121 x 1211 = 1211 x 1121

Notice how all the factors are comprised of the digits 0, 1 and 2 only. If we relax the rule that square numbers are excluded then, with their inclusion, 26962 is a member of OEIS A158642:

A158642

Palindromic numbers which are the product of a number n and its reversal (n written backwards)

The members of this sequence, up to two million, are:

0, 1, 4, 9, 121, 252, 484, 10201, 12321, 14641, 20502, 23632, 26962, 40804, 44944, 1002001, 1113111, 1226221, 1234321, 1357531

The previously mentioned non-square numbers are marked in blue.

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26962 is a palindrome which is 

a product of a pair of emirpimes

In the case of 26962, it can be noted that the factors 122 and 221 are emirpimes:

• 122 = 2 x 61
• 221 = 13 x 17

This property of being a product of emirpimes qualifies 26962 for membership in OEIS A158126:

A158126

Products of emirpimes pairs, sorted.

The initial members of the sequence are:

765, 1612, 3627, 4606, 4930, 26962, 39483, 48763, 58765, 61306, 69723, 85405, 102910, 107485, 118809, 129682, 134458, 136467, 140572, 146047, 148930, 151209, 155038, 162409, 178555, 194242, 196315, 203098, 213310, 236421, 283798, 291247

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26962 is a palindrome 
with four distinct prime factors

26962 has four distinct prime factors: 2, 13, 17 and 61. Palindromes with this property are fairly rare and constitute OEIS A046394: 

A046394

Palindromes with exactly four distinct prime factors.

The initial members of this sequence, up to 100,000, are (permalink):

858, 2002, 2442, 3003, 4774, 5005, 5115, 6666, 10101, 15351, 17871, 22422, 22722, 24242, 26562, 26962, 28482, 35853, 36363, 41314, 43734, 43834, 45654, 47874, 49494, 49794, 49894, 51015, 51315, 51415, 53535, 53835, 53935, 56865, 58485, 59295, 59595, 60006, 62526, 62826, 64246, 64446, 66666, 66766, 68286, 73437, 74347, 78387, 81618, 81718, 83638, 83838, 87078, 87178, 89598, 89698, 92829, 96369, 98889

26962 is also a palindrome in base 11, being represented as 19291. This property qualifies it for membership in OEIS A180454.

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26962 can be written as a sum of two distinct palindromic primes in three different ways

Still on the topic of palindromes, 26962 is a member of OEIS A356854:

A356854

Palindromes that can be written in more than one way as the sum of two distinct palindromic primes.

The primes are 10301 + 16661 = 10601 + 16361 = 11411 + 15551 = 26962.

The initial members of the sequence are (permalink):

282, 484, 858, 888, 21912, 22722, 23832, 24642, 25752, 26662, 26762, 26862, 26962, 27672, 27772, 27872, 27972, 28482, 28782, 28882, 28982, 29692, 29792, 29892, 29992, 40704, 41514, 41614, 41814, 42624, 42824, 42924, 43434, 43734, 43834, 43934, 44744, 44844, 44944, 45354

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26962 is a palindromic Ulam number

26962 is also an Ulam number and so, being a palindromic Ulam number, it qualifies for membership in OEIS A173542:

A173542

Palindromic Ulam numbers.

The initial members of the sequence are:

1, 2, 3, 4, 6, 8, 11, 77, 99, 131, 282, 363, 414, 434, 585, 646, 949, 2112, 2332, 2552, 2662, 5335, 5665, 8008, 8338, 8668, 10501, 13531, 13931, 15251, 16961, 17071, 18381, 18581, 18681, 22122, 22322, 23632, 23932, 25452, 26962, 28582, 28682, 30703, 30803, 32123