Is 8795 a boring number? General forum
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Ingo Althofer at 20080314
Mathematicians con prove that there
are no boring natural numbers. (If there
were some, take the smallest one – and
voila, this one would be interesting at
least because it were the smallest boring
one.)
In real life things are a bit differently
(and being “the smallest boring one” is sort
of second classinteresting). Well known is
“The OnLine Encyclopedia of Integer Sequences”,
directed by Neil Sloane. Its web address is
http://www.research.att.com/~njas/sequences/
Visitors of the site may enter any string in
the search line, and as a result all possible
hits in the data base are shown. Skimming the
encyclopedia shows that
the smallest natural number not present is
8795
Does anybody here has an idea why 8795 might
be interesting (for instance as member of some
“interesting” sequence). Of course, it would be
best not only to have an idea but also to
achieve that 8795 comes in the encyclopedia.
Ingo Althofer (who did something rather
interesting onthe 8795th day of his life). 
wccanard at 20080314
"Mathematicians con prove that there are no boring natural numbers."
I don’t buy that. Isn’t this basically the same line of reasoning as Russell’s Paradox etc etc? If you try and formalise what a proof is, then you can define boring as the smallest number N without a proof that it’s interesting, but the problem is that this will surely not be a proof that N is interesting, because the definition of N will be beyond the scope of what a proof isit’s kind of a “metaproof” only.
8795 looks like a very boring number to me, though. In fact the most interesting thing about it that springs to mind is that it’s so large! I’m sure there are plenty of smaller boring numbers. Ingo: can you explain more carefully what you did? Are you saying that 8795 is the smallest number you found that is not mentioned in the encyclopaedia at all? You have to be careful with arguments like this: there are plenty of very “dense” sequences in the encyclopaedia: for example the encyclopaedia will contain the sequence of positive odd integers, but the entry in the encyclopaedia for sequence won’t mention 8795 because the encyclopaedia only carries the first 40 or so terms of the sequence, and then will give a general formula for the rest of the terms.
If you have a few hours to spare you can try and find the first 40 numbers not mentioned at all in the encyclopaedia and then submit them as a new entry. I think Sloane might have strong reservations about this sequence though :) This might also already have been donedid you search for “boring”?
wcc 
Ingo Althofer at 20080314
Hi wcc,
> "Mathematicians con prove that there are no boring natural numbers."
>
> I don’t buy that.
I meant “proving” in the sense of having
social agreement on this statement.
> 8795 looks like a very boring number to me, though...
> I’m sure there are plenty of smaller boring numbers.
The answer is “no”, when you accept that a natural number
is not boring when it is explicitly mentioned in Sloane’s
encyclopedia.
> .. can you explain more carefully what you did? Are you
> saying that 8795 is the smallest number you found that
> is not mentioned in the encyclopaedia at all?
Right. Lisa Schreiber, one of my math students, made a complete
automatic scan of the database (on January 11, 2008). As a
result, amongst other things she got a count how often each
number was present (as a string) in the database.
8795 was the only natural number below 10,000 which
was not present.
> ... the encyclopaedia only carries the first 40 or so terms
> of the sequence, and then will give a general formula for the
> rest of the terms.
Right. We had searched only for numbers mentioned
explicitly as a string.
> If you have a few hours to spare you can try and find the
> first 40 numbers not mentioned at all in the
> encyclopaedia
We have the first 50 (from date January 11, 2008).
> and then submit them as a new entry. I think Sloane
> might have strong reservations about this sequence though :)did you search
Me, too.
> This might also already have been done
> for “boring”?
Today, “boring” gives five hits – none of them in relation
to “small boring numbers”.
Ingo.
PS: The smallest negative integer not in the database on
January 11, 2008 was 634 (meaning absolute value). 
MichaeI X at 20080314
I do not understand your proof:
In a first step you setup a list of potentially boring numbers, then you remove the smallest one, because it is that “second class interesting” one.
Then you continue with that method, until it becomes boring:
Voila, the rest is a list of really boring numbers.
In practice, you should run both parts of that procedure in parallel or take a nonprocedural approach:
Assumed there were an infinite amount of natural numbers and some really, really boring ones among them. Every eliminating method becomes boring sooner or later, then there are still some boring numbers left.
You noticed well, my counterproof has the assumption that something can “become boring”. That’s what I remember from university:
Mathematic is extremely hard when Dynamics is involved. 
furbolero at 20080314
The proof is supposed to be the next one:
Suppose there is a set of “boring numbers” (“those with no special property”, take it loosely).
However the lowest number IS the “first lowest number” and then it has a special property, and thus is not boring.
Silly reductio ab absurdum :) 
furbolero at 20080314
sorry the typo:
...the lowest number in that set IS the “first boring number”, and then... 
slaapgraag at 20080314
Well, another proof of hw boring this is:
LG player nr 8795 is zargamox, he never played one single game here.... 
MarleysGhost at 20080314
> Then you continue with that method, until it becomes boring:
LOL! Michael X, you have the right insight. I’m bored already. 
wccanard at 20080314
Here’s my favourite example of this “boring” paradox, which actually fits in very nicely in the context of a games site. We don’t really want to play games that go on forever! The nice thing about games like hex and dots and boxes is that they are guaranteed to finish in a finite time, and if we make little modifications to games like chess (say, we demand that if a position repeats 3 times then the game is a draw, rather than saying just that either player may claim a draw) then these games become finite too. It’s obvious what I mean by “finite”, right? A game is finite if every play of the game finishes in a finite time.
OK, so let’s say that on a games site not too far from here, all the games on it are finite (let’s say we have dots and boxes, hex and four in a row). In fact let’s say that one of the guidelines for implementing new games onto the site is that they must be finite games. Now it’s time to introduce a new finite game onto the site. Here’s a fun finite game! It’s called “hypergame”. The rules of it are as follows. Player 1, on his first move, names one of the games on the site; that is his move. And then the two players start playing the game that player 1 has named, with player 2 having the first move. So for example, in a game of hypergame, player 1 can say “dots and boxes”, and then a game of dots and boxes starts with player 2 having the first move. Hypergame is a neat game! You can challenge someone to hypergame and then consider your opponent, and try and find a game that you’re better at than he/she is, and play that game as your first move. Hypergame in fact encourages people to get better at every game on the site! So Hypergame is a Good Thing. Moreover, Hypergame is also clearly a finite game, because on Player 1’s first move he has to name a finite game, and that game will clearly finish in a finite time. So Hypergame is finite.
The site admin likes Hypergame so much that he implements it as one of the games on the site.
A few weeks later he is bewildered by a complaint from a user that it appears that an infinite game is going on on the site, which is contrary to the guidelines. And of course the game is:
P1 "hypergame"
P2 "hypergame"
P1 "hypergame"
...
:) 
Gregorlo at 20080314
wccanard, i’m pretty sure i explained that paradox here in the forums a few years ago ;)

Gregorlo at 20080314
i’m looking for it! unfortunately, i have just discovered google doesn’t throw results of LG other than empathy games :(
so i’m handlooking in all pages in all forums... be patient... i will shallow my words if not ^{_}
it’s been my alltime favourite paradox! i can hardly believe i did not post it here!! :D 
Gregorlo at 20080314
Damn! I surfed in all forums pages since i am registered here, and didn’t found my post. I just read the titles, so maybe i missed the right post. :( i’ll check more tomorrow...
btw, i laughed out loud again reading all those posts of AndrewVillasante, who copypasted a lot of sentences from http://www.7freedom.com/, but changing the order, and sometimes changing some words. Chaotic form of art! I laughed again, how nostalgic!
returning to topic,,, nobody remembers me posting about hypergame?? :( 
Robin at 20080314
Gregorio, I remember that someone posted that paradox a few years ago. If you say it was you, I believe you :)

wccanard at 20080314
Gregorio: I am well aware that the paradox is wellknown :) I certainly didn’t invent it!

Ray Garrison at 20080314
It would seem that 8795 is not boring at all. Just look at all the excitement it is causing in this forum!

Gregorlo at 20080314
wccanard,,, of course i didn’t want to imply that!!! :D
i just wantd to point out that it was “discussed” before in the forums ;)
and i agree with Ray!!! 
wccanard at 20080314
8795 is boring, because if Sloane had decided to store a different number of numbers by default in his encyclopaedia then we would have got another number.

wccanard at 20080314
To amplify my point, search the encyclopaedia for the sequence
95,195,295,395,495
You get a few very dull sequences, including things like the fascinating
"Numbers n such that string 9,5 occurs in the base 10 representation of n but not of n1"
etc etc, and 8795 would be in a sequence like this were the first 100 or so terms of each sequence to be listed rather than just the first 30 or so. If Sloane thinks that a sequence like this is worth including in the encyclopadeia
then why not do the same thing with the string “7,9,5” and voila! 8795 is officially interesting!
Here’s an idea which sounds much more intrinsically interesting (to me at least): instead of doing what you did, which relies crucially on how far people could be bothered to write down their fascinating sequences, why not find the smallest positive integer f(n) which is not mentioned as the m’th term of any sequence in the database for 1<=m<=n? This gets around the problem of 8795 being the 8795’th term in the sequence 1,2,3,4,5,6,... . I might be much more persuded by an argument that f(1) is interestingbecauseitissoboring, although of course there’s a danger that 57 is the first term in the fascinating sequence of numbers that have got a 5 and a 7 in, and not the first term in any other sequence... 
Ingo Althofer at 20080314
Thanks for all the feedback.
I’ll try to give a whole bunch of answers
in this posting.
Michael X wrote:
> Then you continue with that method, until it becomes boring:
> ... You noticed well, my counterproof has the assumption that
> something can “become boring”. That’s what I remember from
> university: Mathematic is extremely hard when Dynamics is involved.
Good point to mention the aspect of dynamics.
************************************************
wccanard wrote about hypergame; ... Player 1, on his first move, names one of the games
> on the site; that is his move. And then the two players
> start playing the game that player 1 has named, with
> player 2 having the first move...
The paradox might be simply repaired by formulating:
The first move of hypergame is to propose any LittleGolem
game EXCEPT hypergame.
**********************************
Ray Garrison:
> It would seem that 8795 is not boring at all.
> Just look at all the excitement it is causing in this forum!
Unfortunately, most of the excitement is not about 8795
itself, but about my introducing paragraph and about
weaknesses of Sloane’s dictionary. Had I forseen this,
I would have simply asked
"Who knows GOOD reasons why 8795 might be interesting."
On the ohter hand, I like the discussion. It is not
boring for me, yet.
********************************************
wcc made several interesting remarks. One of
his central points is: 8795 is not in the data base
mainly by bad luck, because it comes too late in
certain sequences which are in the data base.
I like wcc’s proposal:
> ... why not find the smallest positive integer f(n)
> which is not mentioned as the m’th term of any sequence
> in the database for 1<=m<=n?
When Lisa Schreiber would not have other (more interesting)
duties I would ask her to work out this sequences f(n), for
instance for all n, 1 <= n <= 40.
****************************************
In one of his answers css wrote:
> ... and voila! 8795 is OFFICIALLY [capital letters
> by me] interesting!
Although Sloane’s encyclopedia is the first address
for many research question, it is of course not OFFICIAL.
And, as not only Sloane himself is feeding the monster, but
a whole team (more than 20 people in the editorial board),
it is quite natural that soft or even nonserious entries
will occur in the data base.
Perhaps Sloane should set up a second team of helpers
who only had to try cleaning up?! (But to state it clearly:
The percentage of garbage in Sloane’s eancyclopedia is
far far below the percentage of garbage in google and co.)
*******************************************
Finally, a little exercise on one of wcc’s “extreme” examples:
> To amplify my point, search the encyclopaedia for the
> sequence 95,195,295,395,495
>
> You get a few very dull sequences, including things like the
> fascinating "Numbers n such that string 9,5 occurs in the
> base 10 representation of n but not of n1"
Question: What is the asymptotic density of this set within
the natural numbers?
Ingo. 
wccanard at 20080316
Just because a number is prime or semiprime (a product of two primes) doesn’t make it interesting, does it? Doesn’t it have to be something like “the smallest semiprime which is the sum of 7 cubes in 2 different ways” or something?

berghildur at 20080331
8795 is related to the unlucky number 13 and the number of the Beast 666:
8795 = 13*666 + 666/(6+6+6) + 666/6.66

Let G be a generalised Fibonacci sequence, whose terms G(n) are defined as follows:
G(1) = G(2) = G(3) = 0
G(4) = 1
G(n) = G(n1) + G(n2) + 10*G(n3) + 5*G(n4) {n>4}
The first few terms of this sequence are:
0, 0, 0, 1, 1, 2, 13, 30, 68, 238, 671, 1739, 5130.
This sequence has the following property:
Sum{k=1 to infinity} ( G(k) / 10^k ) = 1/8795.
This sequence happens not to be unique, since – strangely enough – there exists also a generalised Fibonacci sequence U (for “unlucky”) and another one, B (for “Beast”), such that Sum(U(k)/13^k) = 1/8795, and Sum(B(k)/666^k) = 1/8795. The discovery of these sequences is left to the reader.

8795 is also related to the number pi, as follows:
pi/4 = arctan(1/2) + arctan(1/5) + arctan(1/13) + arctan(1/22) + arctan(1/633) + arctan(1/2985) + arctan(1/7614) + arctan(1/8795) 
wccanard at 20080401
Heh :
))
Regarding the first comment, what is the smallest number not “related to 13 and 666”? ;
The second comment is indeed nice. I’ve seen this trick before: it was explained to me in the following way: “explain why 1/9899=0.0001010203050813213455...” (the point being that 1,1,2,3,5,8,13,21,34,55 are the first few terms in the usual Fibonacci sequence).
The 3rd is also niceagain a “standard trick” if you like, but it somehow looks more spectacular than it is! You could try and argue for 2,5,13,22,633,2985,7614,8795 to be added to Sloane’s database ;) 
Ingo Althofer at 20080407
Hi Berghildur,
greetings from Shanghai, where I just read
your contribution on 8795.
> Let G be a generalised Fibonacci sequence, whose terms G(n)
> are defined as follows:
>
> G(1) = G(2) = G(3) = 0
> G(4) = 1
> G(n) = G(n1) + G(n2) + 10*G(n3) + 5*G(n4) {n>4}
So, the generating coefficients for this sequence
are a rather simple finite sequence:
(0, 0, 0, 1; 1, 1, 10, 5)
> The first few terms of this sequence are:
> 0, 0, 0, 1, 1, 2, 13, 30, 68, 238, 671, 1739, 5130.
>
> This sequence has the following property:
>
> Sum{k=1 to infinity} ( G(k) / 10^k ) = 1/8795.
Did someone else check if this result is reallz true?
> This sequence happens not to be unique, since – strangely enough – there
> exists also a generalised Fibonacci sequence U (for “unlucky”) and another
> one, B (for “Beast”), such that Sum(U(k)/13^k) = 1/8795, and
> Sum(B(k)/666^k) = 1/8795. The discovery of these sequences is left to
> the reader.
Are the generating coefficients for these sequences similarily
simple, compared with the first sequence?
Ingo.
PS: I always thought Berghildur was a volcano
hunter from Iceland – and not from Kyrgysistan. 
wccanard at 20080407
> > Sum{k=1 to infinity} ( G(k) / 10^k ) = 1/8795.
> Did someone else check if this result is reallz true?
It’s easy to check. Consider the generating function associated to the recurrence relation; it’s F(x)=sum G(k).x^k. Now use the initial conditions and the definition of the recurrence to write F(x) as a rational function of x [we get F(x)=x^4/(1xx^210x^35x^4) ] and finally substitute in x=1/10 to compute the infinite sum. If you do this exercise you’ll easily see how Berghildur reverseengineered the process and obtained the recurrence relation from the number. The clever observation is not obtaining the recurrence relation, it’s observing that you can rig it so that the coefficients are so small: this boils down to the fact that 8795=10000100010010*105*1. Now you can answer your other questions yourself, armed with the trick. 
Ingo Althofer at 20090204
Finally, 8795 has finished its life as a boring number.
Thanks to Omar E. Pol, it is now present in the
Online Encyclopedia of Integer Sequences. Namely, it
has the form
p(n)*p(n+2)  2*p(n+1),
where p(n) is the nth smallest prime number.
Set p(n)=89, p(n+1)=97, p(n+2)=101.
Cheers, Ingo.
A152532 
Carroll at 20090205
Isn’t p(n) simply the nth prime number? (“http://primes.utm.edu/nthprime/index.php#nth”)? Here for n=24,25,26.
So what is the next “Boring Number” so we may laugh a little more? 
wccanard at 20090205
This whole thread is based on what I believe is a false premise. The original claim was that 8795 was the smallest positive integer not in Sloane’s tables. But 8795 is in sequence A000027, which is the sequence 1,2,3,4,5,6,7,8,9,... .

Hjallti at 20090205
Is this the list of all numbers, or the list of all numbers without double digit?

wccanard at 20090205
In some sense, Hjallti, you’ve put your finger on the problem. Sloane doesn’t list the first 1,000,000 terms of every sequence, he just stops when he gets bored (like I did; A000027 is “all positive integers”). So 8795 is probably in many of the sequences in the database (because there are all manner of sequences in there, e.g. “positive odd numbers”, “numbers with a 5 in” etc etc), it’s just that people got bored before 8795 every time. Probably there are other ways of getting 8795 inyou can “contribute new terms” to Sloane’s database! Perhaps someone wants to contribute a few more numbers that have a 5 in? ;)

Ingo Althofer at 20090205
What I meant:
Before December 06, 2008 8795 was the smallest integer
not mentioned EXPLICITLY in the data base.
@Carroll:
I will try to look it up and tell you.
Ingo. 
Ingo Althofer at 20090206
@Carroll and all other numerologists:
After 8795 has come in, the smallest numbers
not explicitly in the Encyclopedia are:
11630
12067 (special: 12067 is in !)
12407
Cheers, Ingo 
Tim at 20090206
From a visual point of view, 11630 certainly is one of the most boring numbers I’ve ever seen, although its octal equivalent 26556 looks promising ...