10
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What is the next number in this (almost certainly infinite) number sequence?

$23, 2, 1, 11, 37, 4, 7, 15, 8, 66, 48, 31, 202, 84, 49, 173, 23, 42, 167, 69, 147, 97, ?$

(Here's a helpful link for your convenience)

Daily hint #1:

Contrary to what one might think, this sequence wasn’t invented in Philadelphia, but on a small island group off the coast of... Aw phuket, I’m not going to just tell you that, now am I?

Daily hint #2:

To get to the island group, you can take a Boeing B-29 Superfortress, if you can find one that still flies. (They didn't make make those planes in Greece, though.)

Daily hint #3:

Now, if you have figured out the other clues, actually finding the answer should be easy as duck soup. No, I meant it's easy as falling off a log. Actually, not that, either. Easy as pie? No! That doesn't enter the picture at all! Easy as A,B,C? ..Doesn't feel exactly right either. But it's really very easy.

April Fools' Extra Bonus Easter Egg clue:

If you don't yet know, where you should be searching, here's another clue: If you start with five, halve it, write the result in Greek and double it, you are already there. (To find your way back to 5 when you are done, you can take one away and square up what's left.)

Final Hint, a glimpse into the future:

Here's what the sequence looks like later on. The first 90 numbers in the sequence have been omitted in order to show a point of interest:

$... 22, 71, 333, 82, 103, 16, 8, 190, 12, 1446, 1575, 828, 1036, 671, 66, 449, 4119, 1772, 2423, 769...$

(This hint is final only in the sense that I'll post the next hint when anyone asks for one.)

Daily Hint #(Final+1):

If you are feeling a bit irrational about the place where you are searching, don't worry, it's probably quite normal, and anyway, what you want to find are the most natural thing in all mathematics. Just remember to write down where you found each of them, and all will be well.

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    $\begingroup$ Ah-ha! I see that you have fallen into my little trap! Clicking on the link makes the helpfulness disappear, you see! (*cue ridiculous evil cackle*) $\endgroup$ – Bass Mar 29 '18 at 11:43
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    $\begingroup$ Well, I'm pretty sure I understand the hints (and perhaps also the title) but I'm currently still drawing a blank on applying them to the actual question... $\endgroup$ – Gareth McCaughan Mar 31 '18 at 16:20
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    $\begingroup$ Phuket: minced oath, or clue? $\endgroup$ – tox123 Mar 31 '18 at 23:59
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    $\begingroup$ @tox123 both, of course :-) $\endgroup$ – Bass Apr 1 '18 at 6:07
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    $\begingroup$ I'm with Gareth. I see what all the hints are pointing to, but what to do with the result of the hints, I have no idea. $\endgroup$ – tyobrien Apr 2 '18 at 23:04
5
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The next number in the sequence is

? = 2, the 23rd term

The sequence is

The position that the natural numbers (1, 2, 3, ...) can be found in the decimal representation of $1 + \sqrt5$
$1 + \sqrt5$ = 3.23606797749978969640917366873127623544061835961152572427089...
That is, the number 1 occurs at place 23, the number 2 at place 2, 3 at place 1, etc. We are not counting the decimal point as a place.

Explanation

The golden ratio is $\phi = \frac{1 + \sqrt5}2$, so $2\phi = 1 + \sqrt5$. This is alluded to in the clues as follows.

A philosophical puzzle, Philadelphia and the Philippines as mentioned in the title and clue #1.

#2 I'm not sure, reference to the Greek letter but I couldn't work out the significance of B-29

#3 I think this clue is trying to ask us to look for a natural sequence, or give us hints on the alphabet, leading back to phi.

Easter egg clue: Half of five is fi or $\phi$ in Greek, and we are looking for the sequence $2 \phi$ by doubling it. This is the biggest hint here. Also, $(2\phi - 1)^2 = 5$

Final hint: There are some clues in this hint, that is that #9 and #98 both are 8, and #10 and #105 are both 66. Also in the original sequence #1 and #17 are both 23. This clue leads us to finding occurrences in the sequence, since the first digit of these are the same.

Final hint+1: A bit of a red herring, but it tells us to note the place that we find the natural numbers, and $\phi$ is an irrational number. I didn't use normal numbers at all. But the sequence of $2\phi$ would have to be normal in order to contain all natural numbers base 10. Hence the question notes "probably infinite" because there may somehow be a natural number that does not occur at all. I don't exactly know the definitions but I believe $\sqrt5$ is normal, though.

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    $\begingroup$ Some parts to add: #1 The Island group is called Phi Phi off the coast of Phuket, Thailand. #2 There is a surviving B-29 is named FiFi. #3 leading to “easy as 1 2 3”. $\endgroup$ – tyobrien Apr 3 '18 at 4:22
  • $\begingroup$ Well done, that's it exactly! Also, good job, @tyobrien, for finding most of the remaining clues. I tried to choose the particular spots in the sequence so that it would be obvious that there's a jump in magnitude of the numbers when changing from single digits to doubles, and then from double digits to three-digit numbers. $\endgroup$ – Bass Apr 3 '18 at 4:35
  • $\begingroup$ Oh, and the comment about the link's helpfulness disappearing when you click it, that's actually quite literal: I happened to discover that when you request "index.html" from OEIS, (alluding to the fact that the sequence consists of indexes into the target number), their web server responds by redirecting the user to the same address, but without the index.html bit. NO I DON'T THink that was obscure at all! Well, maybe just a little bit. :-) $\endgroup$ – Bass Apr 3 '18 at 4:45
  • $\begingroup$ Ah yes, I see the connection with two phi much more clearly now. $\endgroup$ – Jay Apr 3 '18 at 5:53

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