Mr. B's Superfantastic Mystical Sequence Numbers

For a while now I have been working on numbers that I call “Sequence Numbers” or, on really good days, “Mr. B’s Superfantastic Mystical Sequence Numbers”.  There may be another name for them, but I have not found one and I like my names for them anyways.

I define a sequence number as a natural number whose inverse produced a decimal expansion showing a recognizable sequence of numbers.  This sequence may be listed in the Online Encyclopedia of Integer Sequences (www.oeis.org), but if it is not the sequence has to be easily describable.  Either way, the sequence has to produce at least the first 5 non-zero terms of the sequence.

For example: 9801.  Its inverse produces the following decimal expansion:

1/9801 = 0.0001020304050607080910111213141516171819202122232425262728293031323334353637383940414243444546474849505152535455565758596061626364656667686970717273747576777879808182838485868788899091929394959697990001020304050607080910111213141516171819202…

Notice that this decimal expansion produces the integers from 0 to 97 (written in a two digit form): 00, 01, 02, 03, … , 95, 96, 97.  After 97 the sequence “makes a mistake” and skips 98.

I think it’s kind of cool to have a fraction that knows how to count to 97 without making a mistake. If I want to count higher than 97 I can use the sequence number 998001 which produces a decimal expansion that counts to 997 (written in a 3 digit form). 

1/998001 = 0.
000001002003004005006007008009010011012013014015016017
018019020021022023024025026027028029030031032033034035
036037038039040041042043044045046047048049050051052053
054055056057058059060061062063064065066067068069070071
0720730740750760770780790…

I can continue this and find sequence numbers that can count as high as I want: 9997, 99997, 999997, etc.

I know what you’re thinking though.  “Counting? … Big deal!  I can count without using these sequence numbers. Have you got anything really interesting?

Yes I do!  I have found sequence numbers that can multiply, can do powers (exponents), combinatorics, and a bunch (1 bunch = 1 zillion) of things.  I know there are ways to mathematically generate sequence numbers – but I am still amazed by them, which is why I also use the terms “superfantastic” and “mystical” to describe them.

Consider the Fibonacci sequence.  This is a famous sequence made famous by Leonardo Pisano (better known as Fibonacci).  The sequence begins with two 1s, and after that each new number is calculated by adding the two previous numbers together.  1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.  This is not as easy as counting because for each additional step you have to do a simple addition problem to find the next number.  And it’s kind of aggravating because if you make one mistake then all of the number that follow will be wrong also.

Again, I know what you are thinking (cuz I thought the same thing).  “But Mr. B, there can’t possible a sequence number that can count by Fibonacci numbers!”  But, amazing there is such a sequence number – in fact there are several.  You can pick one depending on how far you want to develop your list of Fibonacci numbers.

Take for example: 999,999,999,998,999,999,999,999.  The fraction 1/999999999998999999999999 will produce all of the Fibonacci numbers (written in a 12 digit format) all the way up to 225,851,433,717.

If you don’t believe me, do the math yourself.  “But Mr. B, this number won’t fit in my calculator!”  But you have the tools you need: a brain, the knowledge of how to do long division, a stack of paper, a pile of pencils, and lots of time.  Well, you can do it that way, but I prefer to use other tools too, like the internet.  No, I’m not suggesting you look it up on the internet (and hope that the website you use is no a hoax, or just plain wrong).  I am saying you can use the internet to perform the calculations.  Seriously, as a teacher of amazing mathematics – I don’t have time the time to grade your paper – the “answer” has 684 digits after the decimal point even if you stop at 225,851,433,717!  I recommend the website: www.wolframalpha.com.  It’s a good tool.  It can save you some time.  And it’s a trusted mathematical website (it will do your calculations accurately).

Now I have to make sure that you understand a few things.  First, I did not wake up one day and decide to find numbers that (when written as a fraction) can count, multiply, or do other amazing things.  And I did not think that it would be amazing if there were numbers that had such a property – and then go try to find it in a math book or even on the internet.

Actually, I stumbled across it on the internet in a blog that is not strictly about mathematics.  I found it at www.futilitycloset.com, operate by a friend of mine (though I’ve never met him) with his wonderful wife, which advertises itself as “an idler’s miscellany of compendious amusements”.

When I first saw it I did not believe it.  I thought to myself “Greg has gone off the deep end on this one.  This can’t possibly be true!”  (see http://www.futilitycloset.com/2012/01/08/math-notes-76/)  Greg showed a fraction (1/998001) whose decimal expansion “counts” up to 997 before makes a mistake.

I thought “This is an obvious hoax.  And nobody will know it’s a hoax, cuz they can’t check it on their calculator.  However, I (with my amazing knowledge) will prove that it is a hoax.”  Unfortunately (or fortunately) I found that it was not a hoax.

Well, that got my curiosity going.  I’ve never seen anything like this before.  I wondered if there were other such numbers that might perform other amazing mathematical feats. 

I did not think “Yeah, that’s a little tiny bit interesting, but my dad said I’d never use it in real life.”  If I had used that excuse, I never would have found other “sequence numbers”.  (Besides, what did my dad know about what I would do, or not do, when I grew up?)

I also did not think “Yeah, but those numbers are way too big.  I’m not going to mess with them.”  There are some mathematicians (including me) that believe that mathematics is interesting, and it just gets more interesting with big numbers.  (And besides – I am actually calculating and working with fractions – all of which are less than 1 – so they are actually “small numbers”.  And, by the way, really small numbers can be interesting too.)

I also did not think “FRACTIONS?  Why does it always have to be fractions?  Can’t we just keep it simple?”  Fractions are not that difficult.  We learn to do them in elementary school.  It’s funny (in a sad way) that I’ve actually had to teach calculus students how to work with fractions so they could complete their calculus assignments.

I also did not think “I could get rich with this.”  Probably cuz I can’t.  That’s OK – few mathematicians ever get rich.  What I did think was “This looks interesting …”, and “I wonder …”.  And I did not let some excuses keep me from doing something that I enjoy.

1/999999999998999999999999 =

0.
000000000000
000000000001
000000000001
000000000002
000000000003
000000000005
000000000008
000000000013
000000000021
000000000034
000000000055
000000000089
000000000144
000000000233
000000000377
000000000610
000000000987
000000001597
000000002584
000000004181
000000006765
000000010946
000000017711
000000028657
000000046368
000000075025
000000121393
000000196418
000000317811
000000514229
000000832040
000001346269
000002178309
000003524578
000005702887
000009227465
000014930352
000024157817
000039088169
000063245986
000102334155
000165580141
000267914296
000433494437
000701408733
001134903170
001836311903
002971215073
004807526976
007778742049
012586269025
020365011074
032951280099
053316291173
086267571272
139583862445
225851433717
365435296162
591286729879...

David