23 Feb 2016
Building
Lists of Power Sequences Using Sequence Numbers
David Brooks
SAWB, BE, MS, MS
What
is Power Sequence?
A power sequence is a list of the powers of
a given integer. For example the power
sequence for the number 2 consists of the terms 2^{0}, 2^{1}, 2^{2},
2^{3}, 2^{4}, etc. These
terms are 1, 2, 4, 8, 16, etc.
But I would like to get a list of many more
terms, and I don’t want to keep multiplying by 2.
What
are Sequence Numbers?
They are integers that have a special
property. When you calculate the decimal
expansion of the inverse of a Sequence Number you get a recognizable number
sequence (many of these sequences are listed in the Online Encyclopedia of
Integer Sequences ( www.OEIS.org )
What kind of mathematics do I need to know
in order to work with Sequence Numbers?
You need to know how to take the inverse of
an integer. The inverse of 123 is “1
over 123” or “1 divided by 123” (1/123).
You also need to know how to do long
division – really long division that you can’t do on your calculator. But don’t worry – you can do it by hand on
paper OR you can get on the internet and go to ( www.wolframalpha.com ) and use this free
“super calculator”. It will take inputs
of about 200 digits, and can provide an output of about 3900 digits.
I would not have been able to do these
calculations without access of the Wolfram Alpha website. And I could not check my answers without the
OEIS website. I would recommend you get
on these websites and play around with them to learn how to use them.
However if you understand how to create the
inverse of an integer, and you understand how to take a fraction and do long
division to get its decimal expansion (how to change a fraction into a
decimal), and you learn how to do
these computations on the internet, then you will have it made.
999,998 is a Sequence Number that will
generate a list of the powers of 2 (2^0, 2^1, 2^2, 2^3, 2^4, etc.), written
in six digit strings.
The inverse of this number is:
1/999998 =
The decimal expansion of this number is:
(I have separated the terms with spaces
to make it easier to read.)
0.
000001 000002 000004
000008 000016 000032
000064 000128 000256
000512 001024 002048
004096 008192 016384
032768 065536 131072
262144 …
The sequence above accurately shows the
first 19 terms. If we want a longer
sequence, we will need to use a larger sequence number. I will show you how to do that later.

Let’s try something more
difficult.
999,999,999,993 is a
Sequence Number that generates a list of the powers of 7 (7^0, 7^1, 7^2, 7^3,
7^4, etc.), with terms written in 12 digit strings.
The inverse of this
sequence number:
1/999999999993 =
The decimal expansion of
this fraction is:
0.
000000000001 000000000007 000000000049 000000000343 000000002401 000000016807 000000117649 000000823543 000005764801 000040353607 000282475249 001977326743 013841287201 096889010407 ...
The first 12 terms of
this sequence are accurately shown above.

But these are two
easy. Let’s try to get up to date and
produce a list of the powers of 2016.
999,999,999,999,999,999,997,984
is a Sequence Number that will generate a list of the powers of 2016 (2016^0,
2016^1, 2016^2, 2016^3, 2016^4, etc.), with terms written in 24 digit
strings.
The inverse of this
Sequence Number is:
1/999999999999999999997984
=
The decimal expansion of
this fraction is:
0.
000000000000000000000001 000000000000000000002016 000000000000000004064256 000000000000008193540096 000000000016518176833536 000000033300644496408576 000067134099304759689216 ...
The first seven terms in
this sequence are accurately shown above.
The Online Encyclopedia of
Integer Sequence does not contain this sequence in their collection.

So how to you build a Sequence Number that
will produce a power sequence, with terms written in strings of any length you
choose?
Well it is easier to do that you might
imagine.
First we start with a 1 and a 0:
10
Then we decide how many digits we want our
terms to be written, and add that many zeros to our number. Suppose we want all of the terms up to 18
digits long. Then after the 10 we attach
000000000000000000 (18 zeros)
10,000,000,000,000,000,000
Next we subtract the number that we want to
calculate the power sequence for.
Suppose we choose 5 so that we can generate a list of the powers of 5 (5^0,
5^1, 5^2, 5^3, 5^4, etc.), then we will subtract 5 from the number shown above:
10,000,000,000,000,000,000 – 5 =
9,999,999,999,999,999,995 will be our new
Sequence Number. It will produce a list
of the powers of 5 up to 18 digits long, written in 19 digit strings. I may even produce some accurate 19
digits So let’s try it and see if I am
right.
The powers of 5 (5^0, 5^1,
5^2, 5^3, 5^4, etc.)
A Sequence Number that
will generate a list of the powers of 5, written in 19 digit strings is:
9,999,999,999,999,999,995
The inverse of this
number is:
1/9999999999999999995 =
And the decimal
expansion of this fraction is:
0.
0000000000000000001 0000000000000000005 0000000000000000025 0000000000000000125 0000000000000000625 0000000000000003125 0000000000000015625 0000000000000078125 0000000000000390625 0000000000001953125 0000000000009765625 0000000000048828125 0000000000244140625 0000000001220703125 0000000006103515625 0000000030517578125 0000000152587890625 0000000762939453125 0000003814697265625 0000019073486328125 0000095367431640625 0000476837158203125 0002384185791015625 0011920928955078125 0059604644775390625 0298023223876953125 1490116119384765625 …
The first 27 terms in
this sequence are accurately shown above.
We got all of the terms up to 18 digits long, and one 19 digit digit
term before this Sequence Number made an error.
If we want more terms
(longer terms) all we have to do is adjust our Sequences Number by adding nines
to the front of our Sequence Number.

One of the things that amazes me about
these numbers is that we did not find many of them until we had the use of
computers. Simply because we could not
perform these operations on a hand calculator, and we were too lazy to do long
division on such large numbers. (I am
included in that bunch not wanting to do the long division by hand.)
The mathematical skills needed to do this
are the ability to find the inverse of a number and to do long division –
really long division.
But since I have access to a computer I can
play with these large numbers and discover their properties.
David