I've referred to the Lychrel numbers before in a couple of earlier posts but they always keep cropping up and a dedicated post will serve to remind of what they are, specifically a set of numbers that do not form a palindrome through the process of reversing and adding their digits. Of course, in base 10 it hasn't been proved that such numbers do not form palindromes somewhere down the iterative track but the first Lychrel number, 196, has been tested to a billion digits and no palindrome has been found. There is a site dedicated to these numbers: http://www.p196.org although it hasn't been updated in many years.
Wikipedia says that "about 80% of all numbers under 10,000 resolve into a palindrome in four or fewer steps. About 90% resolve in seven steps or fewer". The article goes to note that "89 takes an unusually large 24 iterations (the most of any number under 10,000 that is known to resolve into a palindrome) to reach the palindrome 8,813,200,023,188" and "10,911 reaches the palindrome 4668731596684224866951378664 (28 digits) after 55 steps". These statistics are relevant because the number of the day when I'm composing this post - 24636 - is a member of OEIS A065320: 53 'Reverse and Add' steps are needed to reach a palindrome.
The first numbers in this sequence are:
10677, 11667, 12657, 13647, 14637, 15627, 16617, 17607, 20676, 21666, 22656, 23646, 24636, 25626, 26616, 27606, 30675, 31665, 32655, 33645, 34635, 35625, 36615, 37605, 40674, 41664, 42654, 43644, 44634, 45624, 46614, 47604, 50673
The various milestones when a number sets a new record for the number of 'Reverse and Add' steps needed to reach a palindrome are recorded in OEIS A065198. The first few such numbers are 0, 10, 19, 59, 69, 79, 89, 10548, 10677, 10833, 10911, 147996, 150296.
More information can be found on this site: https://www.dcode.fr/lychrel-number. A number is delayed when there a multiple steps before becoming a palindrome. The most delayed known is 1186060307891929990 with 261 iterations.
There are potential Lychrel primes and the first three of these are 691, 887 and 1997. These primes form OEIS A135316:
Here is a list of the initial members:
691, 887, 1997, 3583, 3673, 3853, 3943, 4079, 4259, 4349, 4799, 4889, 5581, 5851, 6257, 6977, 8089, 8179, 8269, 8539, 8629, 8719, 10663, 10883, 11777, 11833, 11867, 11923, 11953, 11959, 12097, 12763, 12823, 13397, 13523, 13553, 13597, 13633
on June 6th 2021
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