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Thanks for the link, though I expected to be RickRolled. IMDb says it got 7.4 stars out of 10, but it doesn't sound interesting enough to read the quotes. (And why doesn't the trailer use upper case to make the URL more memorable?)

However now I think you're actually accusing me of some sort of irrationality. Either that or you're projecting from the human tendency to see patterns where none exist.

In contrast, as I understand the situation, I believe there are different degrees and even kinds of randomness. I'm pretty sure that the digits of pi have a high order of randomness, but my hypothesis is that some irrational numbers are quite different from others in terms of their randomness. Let me try to clarify with two examples of strange irrational numbers:

(1) A more-zeroes number: Start with 0.1. At each iteration, you add a string of zeroes that is one longer than in the previous step, followed by a 1 to separate it. This irrational number would start 0.101001000100001000001... This irrational number is completely determined and shows an obvious pattern (and thus seems to lack randomness) but cannot be represented by any pair of finite integers.

(2) A quick-coverage number: Starting from 1, for each positive integer add all of the strings of that length. (I've used base 10 for the example, though any base will work.) Easiest to do it in order, though you could actually make it shorter (in terms of string coverage) by considering the incidental overlaps (and still shorter with clever sequencing). This irrational number would start with 0.012345678900010203040506... (The shortened version might start 0.012345678900203040506.... where "01" has been removed from the 2-digit strings because it appeared earlier and "02" reuses the second "0" from "00".)

These examples are algorithmic rather than formulaic. The first one cannot be assessed by the kind of metric used in the OEIS sequence I cited ( https://oeis.org/A036903 ), whereas the second one would have extremely low values, far below the statistical predictions if the digits were random.

What is it about pi that makes some people think it is fascinating?

Purely serendipitous, but recently I was actually doing some thinking about pi. Actually I was just using pi because the digits were conveniently available. My first line of analysis led to https://oeis.org/A036903, which begins 32, 606, 8555, 99849, 1369564, 14118312, 166100506, 1816743912, 22445207406, 241641121048, 2512258603207... It's hard to follow their explanation, but the 32 is where the first 0 appears, which is the last 1-digit sequence (base 10), 606 is where the last 2-digit sequence appears for the first time, and so on.

What I was actually looking for was a characterization of the randomness of an irrational number, pi in this case. There is a formula that predicts the values of A036903. It is (10^n)*(ln(10^n)) for (n-1)-digit sequences. There was also a diversion into binary representations and the corresponding sequences and formula (though the binary version of A036903 is apparently not in the OEIS).

I don't think this is meaningful, but I did find it curious that for the case of pi 7 of the first 8 decimal predictions were low, while 5 of the 10 binary predictions were low and 5 were high. At least I can't imagine what meaning those results might have.

After some thought, I would now reword my original question as "What are the characteristics of irrational numbers that come closest to (or farthest from) the predicted values?

I would also make one randomness prediction: The last number that completes each series (for any irrational number) should be random. In other words, it was completely random that 0 was the last digit to appear for the 1-digit sequences, and the same degree of randomness should apply for the last 2-digit value, etc.

(However that did give me a really weird idea... It would be possible for almost all of the four-digit (decimal) sequences to appear before the final 3-digit sequence appeared. It would then require at least 37 more digits to finish the 4-digit sequences. (Actually slightly less than 37 digits is possible if the overlaps were arranged carefully.) I think anything approaching such extremely short gaps would be extremely weird--but still random and should be discovered if enough irrational numbers are studied hard enough.)

My initial reaction to this story was to wonder how irrelevant this is from a real world perspective. The actual universe is not flat. Made me wonder how many decimal places actually apply to reality. I'm guessing that it's a larger number somewhere out between galaxies... Here on earth, probably less than 10 digits of pi are significant, and fewer than that if you were on Mercury.

Seems to make more sense to calculate an irrational number that has some rational relationship to the real world. How about e? I guess that means February 7th should be e day?

Recently I was actually doing some thinking about pi, but it was purely a coincidence and I just used pi because the digits were conveniently available. Eventually led to https://oeis.org/A036903 which begins 32, 606, 8555, 99849, 1369564, 14118312, 166100506, 1816743912, 22445207406, 241641121048, 2512258603207...

What I was actually looking for was a characterization of the randomness of an irrational number, pi in this case. There is a formula that predicts the values of A036903. (It is (10^n)*(ln(10^n)) for n-digit sequences. Or is it for (n-1)-digit sequences? I'd have to check my notes... (Guess why I switched from math to computer science.)) There was also a diversion into binary representations and the corresponding sequences and formula (though the binary version of A036903 is apparently not in the OEIS).

Not meaningful, but I found it curious that for pi 7 of the 8 decimal predictions were low, while 5 of the 10 binary predictions were low and 5 were high. At least I can't imagine what meaning those results might have.

After some thought, I would now reword the question as "What are the characteristics of irrational numbers that come closest to (or farthest from) the predicted values?

I might be an idiot, but you'll have to prove it ;)
His proof is still flawed. It looks like mathematical induction (https://en.wikipedia.org/wiki/Mathematical_induction), but it really only is the base case, and there's no inductive step.
Let's say that 12407, 12887, 13258, 13794 are the first uninteresting numbers, because they don't have any special property, and for example don't appear in https://oeis.org/.
12407 is the first uninteresting number, so let's agree this property makes it interesting. What about 12887? It's still uninteresting, and it cannot be the first uninteresting number, because then, what would happen to 12407? They cannot be both the first uninteresting numbers, can they? You might consider the second uninteresting number to be interesting, but what about the 13794th uninteresting number?

PS: I suppose this argument cannot be settled, because "interesting/uninteresting" isn't properly defined.

That sequence is in the database as http://oeis.org/A047778 since at least 1999. Funny, just one month ago, Neil Sloan asked "the smallest prime in this sequence is 485398038695407. What is the full subsequence of primes?" For the moment, the first is also the only prime known in this sequence.

Yes I'm a good programmer, I work in assembly you insensitive clod!

Like I did mention in my reply, I read the summary too fast, and tought that it was another sequence.

Well, it turns out that the sequence I was thinking about exists: https://oeis.org/A057137/

There! :-p

Simple question, simple answer.

Refer to TFS under "Crossrefs".

tl;dr: yes

Looks like two primes are known when you do reverse concatenation of the first a(n) integers. If you concatenate 37765, 37764, 37763, ..., 3, 2, 1, then you get a prime.

There are a lot of things that started offline, moved online, and are still going strong after decades.
The On-Line Encyclopedia of Integer Sequences comes to mind. Neil Sloane started that 50 years ago.

Yes, that's A000027.

A104175 ()
From the words to the song "Jenny's Letterbox" by Tommy Tutone.

"This giant repository, which celebrated its 50th anniversary last year,"

Also:
" OEIS: Brief History
The sequence database was begun by Neil J. A. Sloane (henceforth, "NJAS") in early 1964 ...."
https://oeis.org/wiki/Welcome#...

The only thing worse than a pedant is a pedantic moron.

OEIS is great. I started working on a project and was able to find some other work done on the same sequence(s)... ,3,5,9,7,15,11,27,... -- my project concerns representing the positive integers in terms of their prime factorization and then examining the properties of various operations on this representation. =)

please check http://oeis.org/

Largest square = sum of squares of divisors of n.

1, 4, 9, 16, 25, 49, 49, 81, 81, 121, 121, 196, 169, 225

You got that wrong, it's 1,2,4,8,16,31

http://oeis.org/search?q=1%2C2%2C4%2C8%2C16%2C31

I looked at the Titan test and wondered if the OEIS were allowable, since it's an online encyclopedia. I threw the integer sequences into it and it had all of them. Two of those integer sequences and the other non-integer sequence pretty much require remembering random math facts, just like the first verbal analogy. A trained mathematician would do quite well on the math section as well, in part because of their training. I agree with the GP that IQ tests do not have the level of resolution to totally order people on intelligence.

Sorry, I should have linked to http://oeis.org/A008578 (although it doesn't mention the year 1912).

Today, 1 isn't a prime. But 1912, it was.

I will admit to not having much knowledge about the history of our pursuit of prime numbers, but I'm at a loss about the page you linked to. I understand where 1 could be considered a prime, but that list also includes numbers 4, 10, 12, 14, etc on that same list of primes. What school of thought would consider those prime? I'm genuinely curious, because I feel like I must be missing something obvious.

Today, 1 isn't a prime. But 1912, it was.