Six Sequences - Numberphile

Which of these number sequences do you like best? Vote at bit.ly/IntegestVote
The extra bit of footage is at: • Tony's Favourite Numbe...
More links & stuff in full description below ↓↓↓
This video features Tony Padilla from the University of Nottingham: / drtonypadilla
Here's each sequence on the OEIS:
Khintchine's constant: oeis.org/A002210
Wieferich primes: oeis.org/A001220
Golomb's sequence: oeis.org/A001462
Largest metadrome in base n: oeis.org/A023811
All 7's: oeis.org/A010727
Wild Numbers: oeis.org/A058883
The Aperiodical: aperiodical.com/
Brown Papers: bit.ly/brownpapers
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[@Majorohminus]

my favorite sequence is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10... its the natural sequence and its perfect. the number in the nth position is n and its the first sequence anyone learns.

[@user-mz7cn9hq8v]

@Adi Septiana 1. It was supposed to be sarcasm 2. This sequence is the base for e

[@aforcemorepowerful]

It's also the decimal expansion of Champernowne's constant

[@TheMagicianLiam]

I agree but in base 12. Sorry

[@hkayakh]

How about that sequence but nth position is -n?

[@sankang9425]

Ah yes, the A000027... My second favorite.

[@m3ntalcas3]

i could tell khinchin's constant was his fave he went on about it much more than the others

[@leo17921]

also cause its more complicated

[@rednecktash]

thats what i thought too even before seeing any other ones

[@thomaskaldahl196]

But why the decimal expansion? Is there anything special about it?

[@olivialuv1]

@@thomaskaldahl196 The decimal is cool bc you get to know the approximate value of this godly self-knowing number, as opposed to just some fraction whose value you can't tell by looking at it

[@thomaskaldahl196]

@@olivialuv1 But what's significant about base 10 as opposed to binary or some other base?

[@christosvoskresye]

It would seem to me that the constants in the continued fraction expansion of Khinchin's constant would be more meaningful than the decimal expansion.

[@RedRad1990]

*Tony Padilla:* "I'm not going to tell you which one I like best" *also Tony Padilla:* proceeds and starts by talking about his fav no hints XD

[@anticorncob6]

One of my favorite number sequences is this: 2, 3, 4, 5, 7, 9, 11, 13, 16, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, etc. It's all the primes, plus all the values of p^(2^n) where p is prime and n is a positive integer. With these numbers, every other number can be written as a unique product of these, without repeats. For instance 99 = 11 * 9, that's the "factorization" and there's no other way to do it.

[@DimaVaulin]

Wow!

[@bagelnine9]

Same.

[@smoorej]

Khinchin’s constant is absolutely mind blowing. That any continued fraction expansion of “almost all” numbers gives you Khinchjn’s constant is just jaw-dropping. Question: is the “almost all” numbers all real numbers except the rationals?

[@vocnus]

It is really incredible and yet ture for ''almost all'' numbers... however it is NOT containing each and every irrational number! For example fi=1.618... or the base of natural exponential e=2.718... are irrational numbers which are not under this rule. The fi's fractional expansion goes this way: [1;1,1,1,...] which is the notation for 1+1/(1+1/(1+1/(1+...))), and with the ''e'' it goes this way: [2;1,2,1,1,4,1,1,6,1,1,8,1,1,10,1,1,12,...] = 2+1/(1+1/(2+1/(1+1/(1+1/(4+...))))). In the first example the geometric mean is constant 1, and in the second it goes to infinity as we deal with more and more terms...

[@Catman_321]

you can show pretty easily that no quadratic irrational number has this property. Since the terms in the continued fraction repeat periodically, it will not converge to an irrational number. Same with numbers where the terms in their cf strictly increase, which diverge to infinity. e is similar to these numbers and doesn't converge to khinchin's constant for similar reasons however most generic irrational numbers do have this property

[@numberphile]

Shout-out to Ireland!

[@goutamboppana961]

??????????

[@Spitzenhund]

Its a nice technique that helps with concentration. We are trained to see brown as a constructive material so writing on it makes us think we are doing more than just writing on paper. The tactile sound and feel of the paper also helps with concentration and I honestly think it sounds nice and prefer it over just normal paper or a white board

[@overwrite_oversweet]

This virol ad said something like 95% of KZbin vid get less than 1000 views and my first reaction was "wow so many KZbin vids get over 1000 views"

[@unecomedy13]

for the wild numbers, just add 0.5.

[@ToxicGLaDOS]

Could you do a video on the Tree function? I've looked up some things about it but it's over my head without intense explanation. Mostly about TREE(3) and how it compares to grahams number and other big numbers.

[@alecbader7433]

You were ahead of your time...

[@asheep7797]

4 years ahead of your time.

[@whatno5090]

My favorite integer sequence is and will forever be the look and say sequence.

[@IMortage]

Numberphile had Conway himself talking about the look and say sequence

[@ckmym]

1 11 21 1211 111221 312211 13112221 1113213211 ...

[@Ensivion]

I love how Tony's collar was popped for most of this.

[@naota3k]

This is the only Numberphile video that went completely over my head.

[@ChristianPerfect]

Thanks for your votes, everyone! Golomb's sequence won the vote, but the only sequence we could fit on the trophy was the Wieferich primes so we said that won instead. Look at the trophy on The Aperiodical, it's magnificent.

[@WilliametcCook]

2:53 But I thought God's Number was 20... James Grime was in your video on it...

[@annevanderbijl3510]

Nice

[@patrickmckinley8739]

A086703. The continued fraction of Levy's constant. Levy's is closely related to Khinchin's. This constant also embodies a property of the continued fraction of almost all numbers - and this sequence is itself a continued fraction. We say "almost all" numbers. Just to expand on this, the exceptions are somewhat intriguing. Any number that is a root of a quadratic does not comply. Also, Euler's number e.

[@Devilogic]

When talking about real numbers, "almost all" is typically defined as "all except for a set of (Lebesgue) measure zero". This is the case here as well. The exceptional set here is in fact uncountable! A subset of it is the uncountable set of all reals with only 1 and 2 in their cont. fraction expansions - the geometric mean will be less than (or =) 2, but Khinchin's constant is >2 Another is the uncountable set of reals with numbers >=3 in their expansions - the geo. mean will be >=3, but K0

[@ZipplyZane]

I had the same problem, and I wasn't using the subtitles. It just sounded like descending to me. I think it's that T at the end of strict being right by the A at the start ascending. It sounds like strict-d-ascending.

[@ajeydevadiga6652]

numbers are just awesome......what a beauty...

[@daniellittlewood8471]

I already knew about khnichin's constant and love it, but golomb's sequence is definitely my favourite!

[@Leadvest]

Could you do a video on A027746? It's a list of n by prime factors.

[@AymanB]

Solomon Golomb ! That is a great name.

[@The214thRabidFangirl]

More often then not this channel does a good job at explaining the math so that I can understand it and how cool it is even though with my basic knowledge. This is not one of those times. I am sure it is awesome, but it is way over my head.

[@Cernoise]

Maybe I misunderstood something, but if the first one is just the sequence of digits in a real number, and the integers in the sequence aren't actually used as numbers, it's not really significant as an integer sequence. It's not s very integ sequence at all, let alone the integest.

[@sdvalen7761]

Levy's constant applies to itself and "almost all" numbers in the same sense that Khinchine's does. It's a related property of continued fractions. I nominate A087602 (its decimal expansion) and A086703 (its continued fraction expansion) as my favorites.

[@br0sRchill]

Im watching the entire series this summer and i cant stop thinking of this lol

[@thoughtyness]

What is the proper notation for the continued fraction at 1:10? If inputting into WolframAlpha, etc how would you correctly write it?

[@user-kh5tv9rb6y]

Probably just explicitly. a0+1/(a1+1/(a2+1/(...)))

[@VeteranVandal]

I knew it was the first one, because I knew he would just be able of holding himself in the sequence he liked the most if it was presented first.

[@Lahbreca]

Why was 67 twice in the wild numbers, if that was just the list of numbers that would result from the operation being done on any given number?

[@synchronizerman]

AH! I understand now. Thank you. Just to make sure then: in base 3, would a number such as 5432 be 543 groups of 3 + 2, or 1631? (and it would continue like that?)

[@fearingfearitself]

You didn't unpack the formula behind all the 7s: 7 x 1^n, where n in the position in the sequence.

[@johannesraspe9699]

Will Khinchin's Constant also work for complex numbers? Or at least their real parts or values..?

[@tommyrjensen]

Sick sequences.

[@imspidermannomore]

I knew Khinchin's constant would be his favourite. That sort of kinky stuff makes all mathematicians salivate.

[@michielr1aert]

Khinchin's constant is gotten by writing numbers in a specific way (continued fraction) - are there other way's of writing numbers, which beget other constants?

[@Bigandrewm]

Would it be possible to see a video on Golumb's Ruler? I can see some clear musical applications, but I'd like to see it from a mathematician's perspective.

[@ericsbuds]

fun video. I really liked it and tony is a good at explaining.

[@kujmous]

In the way the Fibonacci Sequence has values by summing the previous 2 values, do any constants or behaviors surface by increasing the number to 3 or higher? 1, 1, 1, 3, 5, 9, 17, 31...

[@richardgaule9415]

You can't beat a bit of numberphile during the school summer holidays , especially in ireland where it always rains !

[@1234567fe]

Great work numberphile!!!

[@synchronizerman]

That makes absolute sense to me now. Thank you for explaining the concept. On another note, would you know why some people argue that base 12 is more intuitive than base 10?

[@tomlloyd8122]

12 has more factors

[@robo3007]

My favourite: 1, 2, 6, 12, 60, 360 and 2520. The only numbers that have more divisors than every single number apart from itself and up to it's double. These are literally the most divisible numbers can be, seeing as doubling the number adds a new power of two to the factors.

[@skalderman]

is 2520 the maximum how about 5040?

[@robo3007]

@@skalderman 5040 doesn't work. 7560 has more divisors than it and is less than 10080 (2*5040)

[@keyofamajor]

aha, thanks! looking up "almost all" on wikipedia says that there are "a number of specialised uses" of the term, which continues to confuse ._. definitely not as bad as "mathematical concepts named after leonhard euler" though

[@mrholm123]

Could you explane Where the simpons factor was calculated, and how ? Such as the factors 1,4 2,4,2,4,1. Or depending of how many factors you need

[@TheBalfrog]

Wieferich Primes are hard to explain, but the best I can give it to you is by simply showing it, Wieferich Primes we know of, 1093, so p=1093, 2^(p-1) which is 2^(1092) can be divided by 1093, and come out with an integer, whereas if you tried say p=5, (2^4)/5 isn't an integer. Because you can rewrite the conjecture 2^(p-1) = 1, it needs to come out with an integer, to be a Wieferich Prime, hope that sort of helped with understanding it

[@ehhorvath13]

in a nut shell, a base is how you describe the place holder. we generally use base 10, so each place holder is a power of 10: one's place, 10's place, 100's place...or in other words 10^0 place, 10^1 place, 10^2 place. If I want base 5, each place holder would be a power of 5. 5^0, 5^1, 5^2 ect. so if I want to write the number "six" in base 5, I would write 11. meaning, one set of 5^1=5 and one set of 5^0=1 ==> 5+1=6.

[@christophersam1989]

Here he means 'almost all' in the measure theoretic sense, rather than cardinality. Just as the interval [0,1] contains 'almost none' of the numbers in the Cantor set, despite being an uncountable subset. Essentially if you picked a number at random there is probability 1 that it gives Khinchin's constant and probability 0 that it lies in the Cantor set.

[@einsteiner900]

Read the Wikipedia article on continued fractions. All rational numbers have terminating (non-infinite) continued fraction representations. Therefore the geometric mean of their terms does not "approach" anything, it just is a fixed value, which will not be the same as Kinschine's constant.

[@annikapeterson4061]

Can there be a whole video on Fermat's Last Theorem?

[@borntoarun]

Can you do a whole video on Khinchin's constant? Specifically, can you do an example of how a certain number, when you do the continued fraction expansion of it, approaches the constant?

[@ckq]

8 years late but essentially if you pick a random number 0-1, It's continued fraction has a 1/1 - 1/2 = 1/2 chance of being 1, 1/2 -1/3 = 1/6 chance of being 2, 1/12 chance of being 3, etc. So the geometric mean is just 1^(1/2) * 2 ^ (1/6) * 3^(1/12)... n^(1/(n(n+1)) which is that constant

[@karolinachmielewska7691]

Can someone explain how Golomb's sequence moves towards the Golden Ratio as n approaches infinity?

[@Sharaton]

You've also missed that there are n factors in the product. If they all were equal to x the product would be x^n and the exponents would cancel out. When you take limits you have to take the limit of the entire expression, not just parts. (Furthermore, the limit would be of the type infinity^0 which is undefined)

[@NeoUno866]

1,11,21,1211,111221,312211, ... You split it up and describe the previous number, where the next number in the sequence is the description.

[@lol-xs9wz]

Golomb's sequence actually kinda blows my mind.

[@dwarduk2]

Take pi for example. a0 is the integer part of that, so 3. Now take the reciprocal of the fractional part. a1 is the integer part of that, so a1 = 7, giving a geometric mean so far of 4.58. Take the reciprocal of the fractional part of what you currently have, and a2 is the integer part of that (15), and so on. I'm going to reply to this comment with some actual data on this as it applies to pi.

[@DiaStarvy]

A similar sequence would be all p^3^n where p is prime and n is a non-negative integer: 2, 3, 5, 7, 8, 11, 13, 17, 19, 23, 27, 29, ... You can get every positive rational number excluding 1 by multiplying or dividing these numbers and factorisations are unique. For example, 28/5 = 7 * 8 / 2 / 5

[@whauk]

how do you define the percentage? if you have a finite set you can just count out the number of elements with your property and compute the percentage. if you have a countable set you can look at all finite subsets and count the elements with your property in each of them and figure out whether the "limit" exists if you make them larger. however on an uncountable set...? compare lesbesgue-measures in any bounded subset? and then figure out whether a limit exists if you make the subsets "larger"?

[@numberphile]

In the video description there are links to all the sequences, a chance to vote for a winner and other stuff...

[@mr.z111]

Ш vs Щ

[@gmdFrame]

@@mr.z111 Прив

[@frogger9801]

YES! I use oeis all the time! :D Awesome that you guys use it too

[@mrphlip]

The subtitles for the "metadromes" section constantly says "strict descending" instead of "strict ascending"... I guess the transcriber misheard? But that caused a lot of confusion for me trying to figure out what was going on...

[@AMotoVlogger]

@Numberphile look out for a paper with a conjecture on Pi and the "All the Seven's" coming to a computer near you.

[@busTedOaS]

that grows exactly as fast as the busybeaver function. you could however use f(x) = busybeaver(x) * busybeaver(x)

[@synchronizerman]

Could you explain what you mean when you say, "base 1" or "base 1,2,3,4?" (for the metadromes sequence). I'm having difficulty understanding how you arrive at each number in the sequence.

[@vinigretzky97]

When will you finally do a video about the zeta function?

[@MultiPaulinator]

Actually, those are just the powers of 2. A perfect number is a number whose entries in its divisor list -- including 1 but not the number itself -- add up to the number in question. Finding them goes something like this: Iff(sic) 2^(p-1) is prime -- which is only possible, though not guaranteed, when p is prime -- then 2^(p-1)*(2^p-1) is perfect.

[@venkatbabu186]

Infinite fraction is a decimal rotation of digits. As the fraction increase the decimals are insignificant and so reduce to k constant. Two most significant and other reduce fast. 3 is the closest. These kind of things are wave guides. Mostly used for encryption FM and AM.

[@ExtremeMagneticPower]

A continued fraction expansion is basically a decimal turned into a sequence, and any digit is turned into an integer. Let's take pi as an example. 3 is the first digit, so that will be the first number in the expansion sequence: a0. 1 is the first decimal: a1 4 is the second: a2 The sequence would be: {3,1,4,1,5,9, ... } In the above sequence, if I was to take a0 + 1 / a1, it would give me 3.1 If I was to take a0 + 1/ (a1 + 1 / a2), it would give me 3.14 Make sense now?

[@venkateshbabu1504]

What is K. A continuous fraction is supposed to end in a bell curve.

[@shamilakhmadov4314]

i still can't get the Golomb's sequence

[@doublelxp]

I was thinking a more interesting "all the sevens" would be 7 in each of the bases, but it would just be 111, 21, 13, 12, 11, 10, 7, 7, 7, etc.

[@mathmachine4266]

The largest metronome base n is (n^n-n²+n-1)/(n-1)². Special case, n=1, the limit as you go to 1 is 0.

[@CoyMcBob]

Couldn't you start Golomb's sequence with a 0? 1 appears 0 times. it would be: 0, 2, 2, 3, 3, etc.

[@earfolds]

0 would appear one time, so the sequence would be 0, 1, 2, 2, 3, 3...

[@CoyMcBob]

No, there is no zeroth place, and each element tells you about the next element, not the previous. The first element will tell you that there are 0 ones. Thus you know that the second element has to be a two, and so on and so forth.

[@CoyMcBob]

In fact, even more than that, you can just dump zeros wherever you want. Examples: 00000666666777777... is completely valid. So is even 10005555566666 and so on. This sequence only works out if you do it by indexing and not by counting, or you define the range as n >= 1

[@earfolds]

I guess that would work, but then you could even argue the sequence is valid as a string of infinite zeroes. The most interesting one is the one with positive nonzero integers only because zero is trivial.

[@CoyMcBob]

Or, you could just define the sequence as one that every single number in it is referenced to in the sequence. That would also work.

[@Sharaton]

The problem with ten is that it uses the factors 2 and 5 when using 2 and 3 would give you just as many divisors, but you would be able to use those factors more frequently as the factors occur in other numbers more frequently. So every benefit you get from base ten you would get from base six, but they would occur more frequently. There are infinitely many numbers you can choose from that have a given number of divisors, but the smaller ones give you the benefits of those number more often.

[@the_blahhh]

Please do a video on the look-and-say sequence! You start off with some seed like 1, then you say it out loud: "there is one one (1 1)," and so the next term is 11. Then you do it again: "There are two ones," so the next is 21. And then "one two and one one": 1211. It has a lot of unexpected properties and is just downright cool B)

[@ragnkja]

I noticed what Tony's favourite sequence was from how he spoke of it. =)

[@a.gorlovicha9169]

Isn't 3 a Wieferich prime? Because 3^2=9 and 9 divides by 3 (2^3-1)-1

[@ThorHC11]

Видео от A.Gorlovich'a You've got it backwards. it's that p² divides 2^(p-1)-1, not divides by it. for three to work, 3 would have to divide by 9, and it doesn't, because 1/3 is not whole.

[@farstar31]

That's strange, the link to the video with Tony's favorite sequence is broken, perhaps the video has been taken down or something else?

[@Lightning_Lance]

Yeah, noticed the same. The link in the description works though.

[@munsking1]

the forth; take any number, write it out, count the letters for it, write that number out, repeat, 4. dunno what you'd call it but i like it ^^ and it works in english, german, dutch and probably some other languages

[@ax999]

They've used it for a while (since the beginning I think). It's provided by Brady, the person who runs the channel and films the videos.

[@CarlosVazquezNajera]

Can you please explain the Kaprekar number / limit ? Thanks in advance.

[@burk314]

With the composite numbers vs the primes you'd be comparing two countably infinite sets. You need another way to compare the sets, say natural density in which the primes have density zero. So in that sense, yes the composite numbers are the vast majority. You are correct in that we need to be careful about our definitions and "vast majority" is vague, but the phrase 'almost every' in mathematics, oddly enough, has a very precise definition which has little to do with topological density.

[@Azaxaaa]

I'm sure for some of the sequences if he expanded on the explanation then it would be understandable. Fitting that many (i can imagine) complex mathematical sequences with full explanation in a ~14 minute video is near impossible. Good job though and great video. Gives you a lot to think about either way.

[@scotttritten309]

There must be some other exceptions to Khinchin's Constant than just rational numbers. For instance, the Golden Ratio is an irrational number, and the continued fraction expansion for it is an endless series of 1's as the coefficients, meaning that the geometric mean of the coefficients would be 1 for any number of iterations, rather than converging to the Constant as the number of terms in the expansion approaches infinity. Similarly, any irrational number constructed in the same way so that all the coefficients in the fraction expansion have the same value would have a geometric mean equal to that value instead of the the Constant.

[@Lazerblade95]

would you ever consider doing a channel on the biological sciences, e.g. biochemistry and medicine

[@WyllieGamers]

Andrew Wiles proof of Fermat's Last Theorem takes a genius to even understand it. And he used some math that didn't exist in Fermat's time. It has yet to be truly solved in the way that Fermat first thought of.

[@kurtiswithak]

What about this sequence? 1, 11, 21, 1211, 111221, 312211, 13112221... It starts with 1 and the next element "describes" the previous, hence the second number in the sequence is "11" meaning "one 1" (describing the previous number). The third number is then "21" meaning "two 1s", and so on. I'm sure there's some very interesting math regarding this sequence..

[@cecasiahaan6801]

Kurtis Fraser 13112221,413213,21122314,31321314,31123314,13123314,13123314,13123314,loop.

[@MrProfetZ]

Conway did some work on this sequence, I think numberphile even has a video about it

[@d4m4s74]

I don't know what this sequence is called, if it even has a name, but I like it 1, 11, 21, 1211, 1231, 131221, 132221, 133221, 232221, 134211, 14131221, 14132241, etc. Wether this sequence has a name or not, try to find out why this is a sequence.

[@cifla]

can u make video about amplituhedron?

[@johncarpenter3428]

This was explained very well.

[@mayplesurup]

numberphile is an awesome channel!!!!

[@Nathanchooper]

YES! Well done.

[@anticorncob6]

That's interesting. Is there a link to those numbers?

[@adamledger6836]

yep khinchin's would definitely be my favorite of those

[@burk314]

I agree. Khinchin's constant assumes that the aliens would have even invented continued fractions in the first place. While I do find them interesting, continued fractions are kind of a niche concept in number theory and aren't really important to a lot of mathematics. Then Khinchin's constant is a non-obvious derivation from that which doesn't work for common numbers. Compared to pi which pops up in many different areas of mathematics, Khinchin's constant is not that important.

[@jamespfp]

I called his favorite after he described its self-referential completeness. Ascribing divinity to it -- I tend to think of that as sentimentality, but it also gave me a chuckle. Nothing is as charming (at present) as completeness, eh?

[@gtom123]

Thumbs up if you could already feel Tony's excitement when describing the Khintchine's constant :)

[@Lopsidationy]

For Khinchin's constant: WHY, oh why, did they not pick their sequence to be the constant's continued fraction expansion??

[@darkkijin]

I love math... I also love the nerdiness in all of these videos!

[@SlyTy98]

I can't follow some of these videos, but it's knowledge so.....keep it up! Also I want more :D