Every one hundred days, as I track my diurnal age, a palindromic numbered day comes my way. Today is day 26562. Sometimes, like today's number, I create a post dedicated to the palindrome. I didn't do this for 26462 but prior to that I've posted about:
- 26362: Another Special Palindrome on June 6th 2021
- 26262: A Special Palindrome on February 26th 2021
- Palindromic Cyclops Numbers (in particular 26062) on August 10th 2020
In the transition from one millennium to another, the gap increases to 110 days e.g. 25952 to 26062. Other posts relating to palindromes include:
- L-th Order Palindromes on January 24th 2019
- Lycrel Numbers on September 14th 2016
- Remembering Reverse and Add, Palindromes and Trajectories on June 22nd 2016
- 22, Reverse and Add on January 7th 2016
Of course, my next palindromic day 26662 will be spectacular but today I'm focused on the less spectacular 26562. Here are some of its properties:
PROPERTY ONE
A046263 | Largest palindromic substring in \(5^n\). |
26562 makes regular appearances in OEIS A046263, in fact it appears in every 16th term:
1, 5, 5, 5, 6, 5, 6, 8, 9, 9, 656, 828, 414, 22, 515, 757, 878, 939, 26562, 9, 9, 8, 101, 55, 464, 3223, 11611, 969, 252, 626, 515, 656, 696, 44, 26562, 7337, 51915, 75957, 797, 989, 949, 747, 787, 9739379, 86968, 707, 4224, 1001, 929, 646, 26562, 61616, 63336, ...
Here are the powers of \(n\) in which it appears up to 100:
- \(5^{18}\) = 3814697265625
- \(5^{34}\) = 582076609134674072265625
- \(5^{50}\) = 88817841970012523233890533447265625
- \(5^{66}\) = 13552527156068805425093160010874271392822265625
- \(5^{82}\) = 2067951531382569187178521730174907133914530277252197265625
- \(5^{98}\) = 315544362088404722164691426113114491869282574043609201908111572265625
PROPERTY TWO
A046394 | Palindromes with exactly 4 distinct prime factors. |
Here are the initial members and their factorisations:
Palindrome Factorisation858 2 * 3 * 11 * 132002 2 * 7 * 11 * 132442 2 * 3 * 11 * 373003 3 * 7 * 11 * 134774 2 * 7 * 11 * 315005 5 * 7 * 11 * 135115 3 * 5 * 11 * 316666 2 * 3 * 11 * 10110101 3 * 7 * 13 * 3715351 3 * 7 * 17 * 4317871 3 * 7 * 23 * 3722422 2 * 3 * 37 * 10122722 2 * 3 * 7 * 54124242 2 * 17 * 23 * 3126562 2 * 3 * 19 * 23326962 2 * 13 * 17 * 6128482 2 * 3 * 47 * 10135853 3 * 17 * 19 * 3736363 3 * 17 * 23 * 31
PROPERTY THREE
A045960 | Palindromic even lucky numbers. |
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, 40204, 40804
Figure 1 reminds us what even lucky numbers are:
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| Figure 1: source |
PROPERTY FOUR
A317976 | a(n) = 2(a(n-1)+a(n-2)+a(n-3))-a(n-4) for n >= 4, with initial terms 0,0,1,0. |
The terms quickly increase in size and the initial terms are:
0, 0, 1, 0, 2, 6, 15, 46, 132, 380, 1101, 3180, 9190, 26562, 76763, 221850, 641160, 1852984, 5355225, 15476888, 44729034, 129269310, 373595239, 1079710278, 3120420620, 9018182964, 26063032485, 75323561860, 217689133998, 629133273722, 1818228906675, 5254779066930, 15186593360656, 43890069394800, 126844654738097
The generating function for these terms is:$$ \frac{x^2(1 - 2x) }{1 - 2x - 2x^2 - 2x^3 + x^4}$$PROPERTY FIVE
A261924 | Numbers that are the sum of two palindromes of the same length. |
In the case of 26562, there are 21 such palindromic pairs:
- (16561, 10001)
- (16461, 10101)
- (16361, 10201)
- (16261, 10301)
- (16161, 10401)
- (16061, 10501)
- (15551, 11011)
- (15451, 11111)
- (15351, 11211)
- (15251, 11311)
- (15151, 11411)
- (15051, 11511)
- (14541, 12021)
- (14441, 12121)
- (14341, 12221)
- (14241, 12321)
- (14141, 12421)
- (14041, 12521)
- (13531, 13031)
- (13431, 13131)
- (13331, 13231)
The pair (13431, 13131) is of particular interest because its members share no digits in common with their addend 26562.
So the wait is on now for my next palindromic day, 26662, which interestingly falls on Saturday, April 2nd 2022, the day before my 73rd birthday. However, my 73rd Solar Return occurs at 8:34pm on April 2nd. My birthday will thus occur on a Sunday just as on the day I was born. The 666 sequence of numbers will span ten days:
26660, 26661, 26662, 26663 (birthday), 26664, 26665, 26666, 26667, 26668, 26669
While on the subject of palindromes, I came across a tweet that I'd posted on a very special day. See Figure 2. The date was Thursday, February 4th 2010, almost 12 years ago. It's hard to read but on that date I was 22,222 days old.
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Figure 2 |
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