## Saturday, May 23, 2015

### Nerdy Numbers - Type 1 and Type 2

"NERDY numbers", "Mr. B's Super-Duper-Fantastic, Magically Mystical and Wonderfully Mathematical Sequences Numbers" or Simply "Sequence Numbers"

As I have mentioned, I have been working on a class of numbers that have the following properties: (1) they are integers, and (2) when they are raised to the -1 power (reciprocal, inverse, or written as a fraction with a one in the numerator, and the number in the denominator) they produce a decimal expansion that shows a recognizable sequence of integers that is either documented in the Online Encyclopedia of Integer Sequences (www.oeis.org) or can easily be described and recognized by other mathematicians.  I call these numbers "Type 1".

Example: 998001 is a type 1 "NERDY number".  It is an integer, and when I calculate the value of 1/998001 I get the decimal 0.000 001 002 003 004 005 ...  a sequence that show counting from zero (000) to 997 in 3 digit chunks.

When I refer to what I call "type 2" I simply mean that the numerator is not 1, but the fraction cannot be simplified.

In both types I make sure that it produces at least the first five non-zero terms of the sequence, but my goal is go well beyond this limit.  Unfortunately, as the numbers in the sequence get larger they often begin to overlap with each other and confuse this process.  Or the fraction may produce a periodic decimal.  Often I can can re-write a sequence number to produce a longer sequence.

1/998001 produces a sequence that count from 000 to 997, skips, 998, then produces 999 - and then repeats itself.  1/99980001 counts accurately from 0000 to 9997 (in 4 digit chunks, and 1/9999800001 counts accurately from 00000 to 99997 (in 5 digit chunks).  My conjecture is that this sequence can be produced in longer and longer sequences as large as I desire.

1/999998 produces a sequence of the powers of 2, in 6 digit chunks.  But if it does not go as far as I need I can use 1/999999999998 (which produces powers of 2 in 12 digit chunks), or 1/999999999999999998 (which produces powers of two in 18 digit chunks).

1/999997 produces powers of 3, 1/999996 produces, powers of 4, 1/999995 produces powers of 5 ... Can you quess what fraction will produce powers of 237?

The other day I showed you a fraction that produces the Fibonacci sequence.  It can be adjust also to produce larger and larger terms to reach what ever goal I set.  1/999998999999 produces terms up to 6 digits long, and 1/999999999998999999999999 will produce terms that are up to 12 digits long.  Imagine that I can list all of the Fibonacci number up to 12 digits long by making one calculation instead of calculating each new term one at a time.

Other Fibonacci like sequences can be done also - such as Lucan, Pell, Jacobsthal, Tribonacci, Tetranacci, and Tridecanacci sequences or even customized Fibonacci like sequences.  1/997997 produces a Fibonacci like sequence, with terms up to 3 digit long, defined as:

OEIS sequence A015518: A Fibonacci like sequence where a(0) = 0, a(1) = 1, and when n>1 then a(n) = 2*a(n – 1) + 3*a(n – 2).

The largest type 1 sequence number I have found so far is an 84 digit number:
99998600009099963600100099799800300299656800
3002997998001000999636000090999986000001.

It's inverse show a terms, in 6 digit chunks, that calculate C(n, 13) or the number of ways to choose 13 items from a group of "n" items.

My problem is that while this is enjoyable - it is lonely work.  I have only found a few people working on finding these numbers.  So if you are one of those "NERDY" people or know of someone that is one of those "NERDY" people - PLEASE CONTACT ME.  I'd love to share some of my work with you, and see the results of some of your work.  Otherwise I will have to declare myself King of the NERDY, in a kingdom of one.

David