Winery

click to enlarge

Emily (my daughter, taking the photo) is helping out in a (short documentary) movie shoot.

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.

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.

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:

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eight billion eight hundred eight million eight hundred eight thousand two hundred ninety-four

eight billion eight hundred eight million eight hundred eight thousand two hundred ninety-nine

eight billion eight hundred eight million eight hundred eight thousand two hundred ninety-one

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 ninety-six

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

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

eight billion eight hundred eight million eight hundred eight thousand two hundred one

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

eight billion eight hundred eight million eight hundred eight thousand two hundred seventeen

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-eight

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-five

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-four

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-nine

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-one

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-seven

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-six

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

eight billion eight hundred eight million eight hundred eight thousand two hundred seventy-two

eight billion eight hundred eight million eight hundred eight thousand two hundred six

eight billion eight hundred eight million eight hundred eight thousand two hundred sixteen

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-eight

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-five

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-four

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-nine

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

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

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-six

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-three

eight billion eight hundred eight million eight hundred eight thousand two hundred sixty-two

eight billion eight hundred eight million eight hundred eight thousand two hundred ten

eight billion eight hundred eight million eight hundred eight thousand two hundred thirteen

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-eight

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-five

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-four

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-nine

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-one

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

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-six

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-three

eight billion eight hundred eight million eight hundred eight thousand two hundred thirty-two

eight billion eight hundred eight million eight hundred eight thousand two hundred three

eight billion eight hundred eight million eight hundred eight thousand two hundred twelve

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-eight

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-five

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-four

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-nine

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-one

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-seven

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-six

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-three

eight billion eight hundred eight million eight hundred eight thousand two hundred twenty-two

eight billion eight hundred eight million eight hundred eight thousand two hundred two

eight billion eight hundred eight million eight hundred eighteen

...

I have made the primes in the above bold.* A much larger version of the list (102 MB) is here (because we lose the bold, a separate, numerical file of the alphabetically first 43628 primes is here). An indexed decimal integer version of the list (20 MB) is here.

* A couple of two-consecutive primes are shown. The first three-consecutive primes instance:

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-seven