I have today finished examining the Leyland numbers from 1000800 digits to 1000899 digits and have found no PRPs therein. This adds to my previous search of 1000900 digits to 1000999 digits completed 26 Jan 2024 wherein I found three PRPs. At nine to ten months per search, I am not anxious to try another such interval. Yesterday was the ninth (year) anniversary of my very first Leyland-prime find. That's a long time to be obsessed with the task. I'm going to take a breather.

## Friday, October 04, 2024

## Thursday, October 03, 2024

### Éric Angelini (1951-2024)

*two*"signes" in order to make the self-referential closing assertion

*true*. Éric's final

*Cinquante signes*article just misses Michael Branicky's (late Sep. 27) emailed response:

## Friday, September 20, 2024

## Friday, September 13, 2024

### Recycling

One of the pastimes of living where we live is to curtain twitch a credible comprehension of human behaviour, such as it is.

## Thursday, September 05, 2024

### Unknown Havermann

I ran into my older sister's genealogy post on an Unknown Havermann yesterday and I was wondering how it ever came to this. Back in 2010 we had a brief email discussion on the matter (I've made some of my reply-text bold for emphasis):

16 Nov 2010

I read with interest your blog on our family tree. I am amazed that you were able to go back that far. Question: Generation V: Heinrich (living with his mother in 1900) +

[*The Generation-V link is a 2022-updated page. The + is a date-of-death marker. In 2010, I did not yet know that date. I have a Neheim1900 excerpt that lists only some family names.*]

I have a Heinrich who was a child of Sette – born: Jan.30/1901; died: Feb.26/1922 which seems to be at odds with your information that he was a child of Heinrich Havermann and Maria Messelke. Perhaps we are talking about another Heinrich?

17 Nov 2010 reply

**My Heinrich came directly from dad's Stammbuch.** I had scanned a copy of it for my computer but I have since lost that file because of a hard drive failure a couple of years ago. You will have to look at the original (assuming it wasn't thrown out). **The residents-living-in-Neheim-in-1900 list basically confirms the existence of that person. He is old enough to be a Fabrikarbeiter and indeed, because his father died in 1876, he would have been at least 24 years old then.**

18 Nov 2010

Thanks for your email clarifying my inquiry. I do have the "Ahnenpass". This seems to be a booklet that belonged to the relatives in Wattenscheid. Is this the same as the "Stammbuch" that you are referring to. I am willing to share the information with you if you want. Let me know.

18 Nov 2010 reply

[*to part 1*] **Yes**, that sounds like it. **It does have Heinrich as a son of Heinrich, does it not?**

[*to part 2*] No. I have all the information I need.

## Sunday, September 01, 2024

### Goodbye Netflix, hello YouTube Premium

Today is the last day of my long-time (since April 2013) Netflix subscription and the first day of my new YouTube Premium subscription for which I'll be paying $5.64 less per month after my one-month-free trial.

## Thursday, August 29, 2024

### The pore structure of Clostridium perfringens epsilon toxin

[article] BioMoleculePlot3D (ribbons) in Wolfram 14.1: click to enlarge |

## Friday, August 09, 2024

### Alphabetizing the integers

Last month I wrote an article called "Integers, in alphabetical order" that illustrated my then-obsession with the topic. In February 1981, Ross Eckler had an article called "Alphabetizing the integers" (in the magazine Word Ways) that loosely anticipated my exploration.

On page 20 of his article, Eckler notes Philip Cohen's approach to the "Bergerson" problem (find the fixed points for integers 1 to n):

**one** **one** **one** four five five five eight eight

**two** three one four four four five five

two **three** one one one four four

two three six seven one nine

two three six seven one

two three **six** seven

two three six

two three

two

The number of fixed-point counts leads us to OEIS sequence A340671, which provides the actual fixed points in one of the links. Eckler gives us the list-sizes which produce matches for the first twenty integers. I have extended this listing to 100000 integers here. Eckler then notes that 3 appears (I am paraphrasing) in A340671 at positions 13 and 31 and asks for additional values. The third occurrence is at position 201 (then 203, 204, 208, ...). He didn't ask for a first 4, 5, 6, ... appearance as these were clearly beyond his realm of realizability at the time. But here they are:

1 1

2 2

3 13

4 202

5 213

6 2202

7 2213

8 14475

9 233164

10 320200

11 449694

12 2450694

13 4367488

14 4580804

15 4580824

The listing is for the first appearance of 1 to 15, giving the position/index at which it occurs. Michael Branicky found #12 on July 7. I determined #13 to #15 yesterday. The fixed points are added here.

## Tuesday, July 30, 2024

### Fun with digital roots

Éric Angelini's most recent blog is here. I was sufficiently intrigued with his challenge to find larger primes than his 20248751248751248751248751248751248751248751248751 that I gave it a go:

z(41) = 20 grows into z(89), Éric's 50-digit prime.

z(91) = 22 grows into z(280), a 191-digit prime.

z(300) = 35 grows into z(1064), a 766-digit prime.

z(3740) = 238 grows into z(**d**+3737), a **d**-digit (>140000) prime.

Up to this point, we have 151 primes in Z. Their positions/indices are: 2, 3, 4, 20, 21, 23, 26, 29, 31, 34, 37, 38, 40, 89, 280, 281, 284, 287, 290, 291, 293, 296, 299, 1064, 1066, 1073, 1078, 1079, 1081, 1084, 1085, 1144, 1147, 1170, 1171, 1184, 1221, 1262, 1263, 1265, 1268, 1271, 1278, 1280, 1287, 1616, 1617, 1619, 1660, 1665, 1698, 1700, 1703, 1706, 1707, 1712, 1719, 1721, 1724, 1729, 1784, 1787, 1789, 1792, 1897, 1899, 1914, 1919, 1920, 1922, 1965, 1972, 1973, 1978, 1983, 1986, 1993, 1998, 2001, 2022, 2043, 2045, 2064, 2075, 2076, 2097, 2100, 2103, 2104, 2106, 2109, 2112, 2115, 2225, 2242, 2243, 2245, 2248, 2293, 2336, 2338, 2363, 2366, 2369, 2371, 2388, 2393, 2396, 2513, 2516, 2517, 2660, 2693, 2696, 2697, 2699, 2704, 2709, 2724, 2727, 2744, 2747, 2748, 2750, 2795, 2800, 2801, 2803, 2810, 3156, 3183, 3188, 3191, 3204, 3206, 3225, 3290, 3293, 3360, 3427, 3462, 3475, 3477, 3484, 3485, 3487, 3506, 3511, 3512, 3515, 3738. So we are looking for prime #152 (the next one). I've put an indexed file of Z (to 3752) here.

One might suppose that in the absence of a definite **d**, we cannot continue Z. Actually, assuming that **d** exists, we can. What we cannot do is assign indices to the continuation, unless one is ok with:

**d**+3738 239 (prime #153)**d**+3739 240**d**+3740 241 (prime #154)**d**+3741 242**d**+3742 2428**d**+3743 24287

...

I've computed 11288 terms of this continuation but, in order to make the file size smaller, I have removed **d**+7693 to **d**+15024 from view. The file is here. The final term at the bottom (index **d**+15025) is prime #190, a 7348-digit prime.

### Festival!

*****: "Maggie's Farm"

*****minus Paul Butterfield, plus Al Kooper & Barry Goldberg

## Saturday, July 20, 2024

### Winery

## Wednesday, July 17, 2024

### Power interruption

One could say that yesterday morning's rainfall was significant! At around 12:40 pm we lost power and, unlike to the usual once-a-week (or so) off-and-ons, this one lasted until 6:25 pm. Apart from having to rediscover some camping survival skills during the outage, restoring my ongoing Leyland prime searches on sixteen computers (once power was restored) took me to 10:30 pm. Unfortunately, seven days of an expected fifteen-day run calculating five million terms of the Philip Cohen sequence is lost.

## Monday, July 15, 2024

### Khinchin geometric means

I first did a geometric-means analysis for Khinchin's constant (K) in early 2000 using 110000 decimal digits of the number. Late Friday, Neil Sloane informed me that a Carlos Simó had calculated one million digits! Converting these to 969680 continued-fraction terms, I did a reanalysis:

click to enlarge |

In this range, the final time that the mean changes sign is from (540114, -9.67033*10^-7) to (540115, 0.0000323051). The record (proximity to K) mean at 353014 is -4.14662*10^-9.

## Thursday, July 11, 2024

### Integers, in alphabetical order

I have been preoccupied, since the beginning of July, about integers in alphabetical order. Claudio Meller wrote this preamble to the Sequence Fanatics discussion list (taken from a British game show called The 1% Club): "If all the numbers from 1 to 20 are written out in words and put into numerical order and then reorganised into alphabetical order, which is the only number that stays in the same position?" Claudio had noticed that, replacing 20 with other integers up to 1000, "all the numbers up to 399 are kept in their position at least once". I replied: "Not 8 or 18". Well, Claudio was wrong about 8 but I was wrong about 18. Claudio pointed out that 18 shows up for 1 to 815.

It hadn't shown up for 1 to 815 for me because I was using a letter-by-letter alphabetical order (spaces and punctuation are ignored) while Claudio was using a word-by-word alphabetical order (spaces are not ignored, a space preceding any letter of the alphabet).

In a Wikipedia article on alphabetical order, the treatment of multi-word strings reiterates again the two-fold situation: word-by-word or letter-by-letter. What's new to me is that the latter may sometimes be called "dictionary order".

Mathematica has an AlphabeticSort function that appears to function word-by-word. I am going to assume that word-by-word is the alphabetical-order prevailing standard and, if anyone prefers letter-by-letter, they ought to spell it out.

We are looking for the start of the integers in alphabetical order (American English, by which we mean Mathematica's IntegerName[#,"Words"]). I will point out that IntegerName uses a non-standard hyphen (character code 8208) instead of the normal, generic hyphen (character code 45). I'll never understand why they did that, but I digress.

In American English all integer names start with one of e, f, n, o, s, or t, so the first term is necessarily "eight". Since we are working in alphabetical order (i.e., word-by-word), the next term should incorporate an additional word. Mathematica has integer names up to (but not including) 10^306 and the alphabetically first word that fits is "billion", so "eight billion" is our second term. As before, we are looking for an additional word that will turn "eight billion" into a new integer. Our third term will be "eight billion eight". Our fourth term will be "eight billion eight hundred".

I decided to Google my four terms and got one hit (the reason I didn't link to the hit is because the webpage doesn't have the Google snippet information on it):

For some reason somebody thought "eight billion eight million" was the fifth term, followed by "eight billion eight thousand", "eight billion eighteen". In my mind, extending the sequence with additional words is the way to proceed. If an additional word is not logically possible, then we consider replacing the final existing word:

**eight billion eight hundred eight million eight hundred eight thousand eight hundred eighty-nine**

**eight billion eight hundred eight million eight hundred eight thousand eight hundred forty-three**

**eight billion eight hundred eight million eight hundred eight thousand eight hundred seven**

**eight billion eight hundred eight million eight hundred eight thousand fifty-one**

**eight billion eight hundred eight million eight hundred eight thousand five hundred eighty-three**

**eight billion eight hundred eight million eight hundred eight thousand five hundred forty-three**

**eight billion eight hundred eight million eight hundred eight thousand five hundred seven**

**eight billion eight hundred eight million eight hundred eight thousand five hundred seventy-nine**

**eight billion eight hundred eight million eight hundred eight thousand four hundred eighty-seven**

**eight billion eight hundred eight million eight hundred eight thousand four hundred seventy-one**

**eight billion eight hundred eight million eight hundred eight thousand four hundred sixty-nine**

**eight billion eight hundred eight million eight hundred eight thousand four hundred thirty-nine**

**eight billion eight hundred eight million eight hundred eight thousand four hundred three**

**eight billion eight hundred eight million eight hundred eight thousand four hundred twenty-seven**

**eight billion eight hundred eight million eight hundred eight thousand nine hundred seven**

**eight billion eight hundred eight million eight hundred eight thousand nine hundred thirteen**

**eight billion eight hundred eight million eight hundred eight thousand ninety-three**

**eight billion eight hundred eight million eight hundred eight thousand one hundred eighty-one**

**eight billion eight hundred eight million eight hundred eight thousand one hundred ninety-nine**

**eight billion eight hundred eight million eight hundred eight thousand one hundred sixty-three**

**eight billion eight hundred eight million eight hundred eight thousand seven hundred eighty-seven**

**eight billion eight hundred eight million eight hundred eight thousand seven hundred eleven**

**eight billion eight hundred eight million eight hundred eight thousand seven hundred fifty-three**

**eight billion eight hundred eight million eight hundred eight thousand seven hundred forty-one**

**eight billion eight hundred eight million eight hundred eight thousand seven hundred sixty-three**

**eight billion eight hundred eight million eight hundred eight thousand seven hundred thirty-nine**

**eight billion eight hundred eight million eight hundred eight thousand six hundred eighty-one**

**eight billion eight hundred eight million eight hundred eight thousand six hundred forty-nine**

**eight billion eight hundred eight million eight hundred eight thousand six hundred ninety-nine**

**eight billion eight hundred eight million eight hundred eight thousand six hundred seven**

**eight billion eight hundred eight million eight hundred eight thousand sixty-one**

**eight billion eight hundred eight million eight hundred eight thousand sixty-seven**

**eight billion eight hundred eight million eight hundred eight thousand thirty-seven**

**eight billion eight hundred eight million eight hundred eight thousand three hundred ninety-one**

**eight billion eight hundred eight million eight hundred eight thousand three hundred ninety-seven**

**eight billion eight hundred eight million eight hundred eight thousand three hundred seventy-three**

**eight billion eight hundred eight million eight hundred eight thousand three hundred sixty-nine**

**eight billion eight hundred eight million eight hundred eight thousand three hundred thirteen**

**eight billion eight hundred eight million eight hundred eight thousand two hundred fifty-nine**

**eight billion eight hundred eight million eight hundred eight thousand two hundred ninety-seven**

**eight billion eight hundred eight million eight hundred eight thousand two hundred seven**

**eight billion eight hundred eight million eight hundred eighteen thousand five hundred sixty-nine**

eight billion eight hundred eight million eight hundred eighteen thousand five hundred sixty-one

eight billion eight hundred eight million eight hundred eighteen thousand five hundred sixty-one

**eight billion eight hundred eight million eight hundred eighteen thousand five hundred sixty-seven**