What's Special About 26862?

As my diurnal age today is 26862, I thought it worthy of some detailed analysis. Recently I've started to give such palindromic days posts of their own. For example:

• Palindromic Cyclops Numbers (this is about 26062 in particular)
• 26262: A Special Palindrome
• 26362: Another Special Palindrome
• 26562: A Mid-Millennial Palindrome
• 26662 (this corresponded to my 73rd birthday)

I've also made several posts about palindromes in general:

• Numbers As Sums Of Palindromes
• Remembering Reverse and Add, Palindromes and Trajectories
• L-th Order Palindromes
• Lycrel Numbers
• 22, Reverse and Add

So let's get started on 26862. To begin with it's what I call a five digit “balanced” number. By this I mean a number such that the sum of the first two digits equals the middle digit and the sum of the last two digits equals the middle digit. There are 330 numbers with this property but only 45 of them are palindromes (permalink). The palindromes are:

10101, 11211, 12321, 13431, 14541, 15651, 16761, 17871, 18981, 20202, 21312, 22422, 23532, 24642, 25752, 26862, 27972, 30303, 31413, 32523, 33633, 34743, 35853, 36963, 40404, 41514, 42624, 43734, 44844, 45954, 50505, 51615, 52725, 53835, 54945, 60606, 61716, 62826, 63936, 70707, 71817, 72927, 80808, 81918, 90909

Thus in the case of 26862 we have:$$ \underbrace{2 \, 6}_{2+6=8} \, 8 \, \underbrace{6 \, 2}_{6+2=8}$$The palindromes in particular that have 8 as the central digit are:

• 17871
• 26862
• 35853
• 44844
• 53853
• 62862
• 71871
• 80808

All eight palindromes and thus linked to the famous 888.

About 90% of numbers can be expressed as sum of two palindromes and 26862 is such a number. It can be represented as a sum of two distinct palindromes in 31 different ways (permalink). If the two palindromes don't need to be unique then we can add 13431+13431 for a total of 32. The palidromes are:

[10001+16861], [10101+16761], [10201+16661], [10301+16561], [10401+16461], [10501+16361], [10601+16261], [10701+16161], [10801+16061], [11011+15851], [11111+15751], [11211+15651], [11311+15551], [11411+15451], [11511+15351], [11611+15251], [11711+15151], [11811+15051], [12021+14841], [12121+14741], [12221+14641], [12321+14541], [12421+14441], [12521+14341], [12621+14241], [12721+14141], [12821+14041], [13031+13831], [13131+13731], [13231+13631], [13331+13531], [13431+13431]

However, of these 31, there are only four pairs in which both numbers are prime (permalink). These are:

[10301+16561], [10501+16361], [11311+15551], [11411+15451]

This property of the number qualifies it for membership in OEIS A356854:

A356854

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

Here are is the list of sequence members up to 40,000:

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

All numbers can be represented as a sum of three palindromes and there are 190 ways to do so with 26862. I won't list them all here but one example is 161 + 949 + 25752.

The number 26862 is not only symmetric internally but also externally in a number of ways. To begin with it is sandwiched between two primes and is thus the average of the two:$$\underbrace{26861}_{\text{prime}} \, 26862 \, \underbrace{26863}_{\text{prime}}$$Furthermore, it is also a practical number that is the average of the previous practical number (two below it) and the next practical number (two above it). Practical numbers are always even. Thus we have:$$\underbrace{26860}_{\text{practical}} \, \underbrace{26862}_{\text{practical}} \, \underbrace{26864}_{\text{practical}} $$These prime number and practical number properties qualify 26862 for membership in OEIS A209236:

A209236

List of integers m>0 with m-1 and m+1 both prime, and m-2, m, m+2 all practical.

Such numbers are few and far between. Here is the list of sequence members up to 100,000:

4, 6, 18, 30, 198, 462, 1482, 2550, 3330, 4422, 9042, 11778, 26862, 38610, 47058, 60258, 62130, 65538, 69498, 79902, 96222

Even triples of practical numbers are infrequent as can be seen from the initial sequence members of OEIS A287682:

A287682

Triples of practical numbers: numbers n such that n-2, n, n+2 are all practical numbers.

Here are the members up to 40,000:

4, 6, 18, 30, 198, 306, 462, 702, 1482, 2550, 3330, 4422, 5778, 6102, 6498, 9042, 11178, 11778, 14418, 15498, 17298, 17442, 19458, 20862, 21582, 22878, 23322, 23550, 25230, 26622, 26862, 26910, 27378, 30210, 34542, 36738, 38610, 39006, 39102

So these are just a few ways in which 26862 is special and their combination of course makes the number unique.