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…

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…

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...

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

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