# Is there a name for this series in pi?

If you read out the sequential digits of prime in such a way that you concatenate repeated digits until they make up a unique number, then presumably after an infinitely long process, you’ll create the set of cardinal numbers. To clarify:

3,1,4,15,9,2,6,5,35,8,97,93,23,84,62,64,33,83,27,95,028,841,971,69,39,937,51,058,20,974,…

At this stage (58 digits of pi), ‘7’ as a single digit hasn’t appeared yet, although 974 has. There’s 30 unique digits at this point.

Is there a name for this series? And if so, are there any interesting qualities of it, or generalisations that can be stated? Any limits on the proportion of cardinal numbers to the pi digits themselves?

I’m sure that the properties of pi have been researched to such an extent that I won’t be able to claim credit for naming the Ferris series, but that’s ok. Always nice to start the new year with an effort at profound thought!

Wow, who would have thought. Are we at a “Simpsons already did it” point in the world of integer sequences?

Well, 4 had been added to the db today at the time of this post. And it’s a holiday. During a school break. We’re not going to run out of new sequences anytime soon.

Looks like that sequence handles zeroes slightly differently than the OP does.

First off, many thanks Dr Strangelove for the link. I’m not that surprised or disappointed to learn that there’s an encyclopaedia for integer series, even though I never thought to search for one.

I had a look at the graph link from there and the logarithmic plot in particular looked rather intriguing. Anyone want to have a stab at explaining the pattern in layman’s terms?

Look’s like an inverse of Benford’s Law. The darkest parts approach, but don’t quite reach, powers of 10. So 9-prefix numbers occur most common, 8 2nd, etc. Not sure why.

There are two features here. We see all numbers with the same number of digits in a horizontal band, darkening toward the top. Then we see a sharp drop in density as an extra digit is added.

Within a band, the darkening toward the top is an artefact of the log scale. Consider the band of 3-digit numbers from 100-999. There are 100 numbers to plot between 100-199, and 100 numbers to plot between 900-999. But on the y-axis log scale, 100-199 is represented by a longer distance than 900-999.

Now, why is there a sharp drop in density between each band? That’s because you are finding numbers in increasing order, and it takes much longer to find a number with one additional digit. So, all number between 100-999 take about the same time to occur, but 1000 takes a lot longer. So you see a sharp drop from 3 digits to 4 digits at 999/1,000, and similarly at 9,999/10,000 etc.

ETA: You would see this in random numbers. If there are unusual features for pi, I think they might be swamped out and not visible. (Not that I expect pi to be anything but apparently random.)

Neil Sloane has been working on the encyclopedia of integer sequences since 1964. I remember consulting it in the 1970’s using an automated email system. As I recall, you typed a comma separated list of sequential elements, and you would get a return email that told you about any matches. There were other rules for submitting sequences of your own.

I attended a talk once by Neal’s colleague, the great John Horton Conway. The subject was, what makes an interesting sequence? So far as I remember, he made no statements about utility. Briefly, he said a sequence was interesting if there was 1) a simple rule for generating successive elements, and 2) theorems about the elements that were simple to state and difficult to prove.

This sequence clearly fits the first criterion, but whether or not the second one fits remains to be determined.