Engel Expansions

I've encountered Engel expansions before and today I was reminded of them again when my day count number, 24648, featured in OEIS A068379 as the Engel expansion of sinh(1/2). The initial sequence of numbers is:

1, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648, 25920, 27224

An Engel expansion is explained by Wikipedia as:

The algorithm for calculating the terms in an Engels expansion is as follows:

This is straightforward enough and I set up a worksheet in Excel to calculate the terms of the Engel expansion for whatever number I entered. I tested it out and all seemed well until I looked more closely at the terms I got for sinh(1/2). Here they are as reported by the worksheet:

The first six terms match the OEIS listing but the seventh diverges by one (623 as opposed to 624) and after that things rapidly fall apart as can be seen by comparing terms. I guess the slight errors that arise as the increasingly smaller u-th terms are divided into one quickly compound and spell disaster. Interesting illustration of the limitations of spreadsheets when very small numbers are concerned.

ADDENDUM:

It's now 1st May 2019 and I've been using SageMath for quite some time now. Here is the SageMath code to generate the Engels expansion of sinh(1/2) up to 24648 (permalink to SageMathCell):

[1, 2, 24, 80, 168, 288, 440, 624, 840, 1088, 1368, 1680, 2024, 2400, 2808, 3248, 3720, 4224, 4760, 5328, 5928, 6560, 7224, 7920, 8648, 9408, 10200, 11024, 11880, 12768, 13688, 14640, 15624, 16640, 17688, 18768, 19880, 21024, 22200, 23408, 24648]

Greek Letters

Given the importance and prevalence of Greek letters in Mathematics, I thought it high time that I make a post to summarise information on this topic. Here is a list of Greek letters and their names taken from Wikipedia:

Click image to see it more clearly

In symbolab (a recent discovery described in a post to my Pedagogical Posturing blog), these are the ways the letters appear:

Some mathematical functions that use these letters and that I'm familiar with or at least heard about are:

The screenshot above was taken from my versal site and here is a list of the ASCIIMathML that was used to create it:

Click image to see it more clearly

Lycrel Numbers

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:

A135316

Primes in A023108(n); or Lychrel primes.

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