Building Lists of Power Sequences Using Sequence Numbers

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 2⁰, 2¹, 2², 2³, 2⁴, 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. If you would like to check the accuracy of these terms you can compare it with the Online Encyclopedia of Integer Sequences: http://oeis.org/A000079 and http://oeis.org/A000079/b000079.txt.

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. If you want to check the accuracy of these terms you can check the Online Encyclopedia of Integer Sequences: http://oeis.org/A000420 and http://oeis.org/A000420/b000420.txt.

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. You can compare these results with the Online Encyclopedia of Integer Sequences at: http://oeis.org/A000351 and http://oeis.org/A000351/b000351.txt.

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