Sequence Numbers: Changing the Length of
the Strings that the Decimal Expansion is Written with.
Counting Sequences:
The Sequence Number 81 will produce single
digit strings from zero to seven:
1/81 = 0.0123456790123…
The Sequence Number 9801 will produce two
digit strings from zero to 97:
1/9801 = 0.000102030405060708091011112…
To continue lengthening the strings keep
adding nines in front of the eight, and zeros in front of the one. The length of the strings will be the same as
the number of eight and nines, or the number of zeros and ones. You should only have 1 eight and 1 one. The number of nines should always be the same
as the number of zeros.
Example: In order to produce strings that
are 100 digits long you will need 99 nines, one eight, 99 zeros and 1 one.
Multiplication Sequences or Multiplication
Tables:
Sequence Numbers for multiplication
sequences must be one of the Sequence Numbers for counting sequences (81,
9,801, 998,001, etc.) divided by the number you want your multiplication
sequence to multiply by (provided the numerator is divisible by the
denominator).
All of the numbers used for the numerators
are divisible by 3, 9, and 27 so any of the counting Sequence Numbers can be
used to show multiples of three. Each
string will be written in ½ of the number of digits in the numerator.
Listed below are the counting Sequence
Number, and their factors.
Let’s look as some of the other factors
first:
Counting Sequence Numbers

Prime Factors

9,801

3^{4} * 11^{2}

998,001

3^{6} * 37^{2}

99,980,001

3^{4} * 11^{2} * 101^{2}

9,999,800,001

3^{4} * 41^{2} * 271^{2}

999,998,000,001

3^{6} * 7^{2} * 11^{2}
* 13^{2} * 37^{2}

99,999,980,000,001

3^{4} * 239^{2} * 4649^{2}

9,999,999,800,000,001

3^^{4} * 11^^{2} * 73^{2}
* 101^{2} * 137^{2}

999,999,998,000,000,001

3^{8} * 37^{2} * 333,667^{2}

99,999,999,980,000,000,001

3^{2} * 11^{2} * 41^{2}
* 271^{2} * 9091^{2}

9,999,999,999,800,000,000,001

3^{4} * 21649^{2} *
513,239^{2}

999,999,999,998,000,000,000,
001 
3^^{6} * 7^{2} * 11^{2}
* 13^{2} * 37^{2} * 101^{2} * 9901^{2}

99,999,999,999,980,000,000,
000,001 
3^{4} * 53^{2} * 79^{2}
* 265,371,653^{2}

9,999,999,999,999,800,000,
000,000,001 
3^{4} * 11^{2} * 239^{2}
* 4649^{2} * 909,091^{2}

999,999,999,999,998,000,000,
000,000,001 
3^{6} * 31^{2} * 37^{2}
* 41^{2} * 271^{2} * 2,906,161^{2}

99,999,999,999,999,980,000,
000,000,000,001 
3^{4} * 11^{2} * 17^{2}
* 73^{2} * 101^{2} * 137^{2} * 5,882,353^{2}

9,999,999,999,999,999,800,000,
000,000,000,001 
3^{4} * 2,071,723^{2} *
5,363,222,357^{2}

999,999,999,999,999,998,000,
000,000,000,000,001 
3^{8} * 7^{2} * 11^{2}
* 13^{2} * 19^{2} * 37^{2} * 52579^{2} *
333,667^{2}

99,999,999,999,999,999,980,
000,000,000,000,000,001 
3^{4} * 1,111,111,111,111,111,111^{2}

9,999,999,999,999,999,
999,800,000,000,000,000, 000,001 
3^{4} * 11^{2} * 41^{2}
* 101^{2} * 271^{2} * 3541^{2} * 9091^{2} *
27961^{2}

999,999,999,999,999,999,
998,000,000,000,000,000, 000,001 
3^{6} * 37^{2} * 43^{2}
* 239^{2} * 1933^{2} * 4649^{2} * 10,838,689^{2}

99,999,999,999,999,999,
999,980,000,000,000,000, 000,000, 001 
3^{4} * 11^{4} * 23^{2}
* 4093^{2} * 8779^{2} * 21649^{2} * 513,239^{2}

9,999,999,999,999,999,
999,999,800,000,000,000, 000,000,000, 001 
3^{4} *
11,111,111,111,111,111,111, 111^{2}

999,999,999,999,999,999,
999,998,000,000,000,000, 000,000, 000,001 
3^{6} * 7^{2} * 11^{2}
* 13^{2} * 37^{2} * 73^{2} * 101^{2} * 137^{2}
* 9901^{2} * 99,990,001^{2}

99,999,999,999,999,999,
999,999,980,000,000,000, 000,000, 000,000,001 
3^{4} * 41^{2} * 271^{2}
* 21401^{2} * 25601^{2} * 182,521,213,001^{2}

9,999,999,999,999,999,
999, 999,999,800,000,000, 000,000,000,000,000,001 
3^{4} * 11^{2} * 53^{2}
* 79^{2} * 859^{2} * 265371653^{2} * 1058313049^{2}

999,999,999,999,999,999,
999,999,998,000,000,000, 000,000,000,000,000,001 
3^{10} * 37^{2} * 757^{2}
* 333667^{2} * 440334654777631^{2}

99,999,999,999,999,
999,999,999,999,980, 000,000,000,000,000, 000,000,000,001 
3^{4} * 11^{2} * 29^{2}
* 101^{2} * 239^{2} * 281^{2} * 4649^{2} *
909091^{2} * 121499449^{2}

9,999,999,999,999,999,
999,999,999,999,800,000, 000,000,000,000,000,000, 000,001 
3^{4} * 3191^{2} * 16763^{2}
* 43037^{2} * 62003^{2} * 77843839397^{2}

999,999,999,999,999,
999,999,999,999,998, 000,000,000,000,000, 000,000,000,000,001 
3^{6} * 7^{2} * 11^{2}
* 13^{2} * 31^{2} * 37^{2} * 41^{2} * 211^{2}
* 241^{2} * 271^{2} * 2161^{2} * 9091^{2} *
2906161^{2}

Notice that 9,801 cannot be used to produce
a Sequence Number showing the multiples of seven because it is not divisible by
seven.
999,998,000,001 can be
used to produce a Sequence Number showing multiples of seven because 999,998,000,001
is divisible by seven.
999,998,000,001/7 = 142,856,857,143
so 142,856,857,143 is one possible sequence number that will produce a decimal
expansion showing multiples of 7 (in this case the terms will be written in six
digit strings).
999,999,999,998,000,000,000,001/13
will produce the smallest Sequence Number that produces a list of the multiples
of 13, because 999,999,999,998,000,000,000,001 is the smallest of the counting
Sequence Number that is divisible by 13.
By the way, 999,999,999,998,000,000,000,001/13 is 76923,076,922,923,076,923,077,
which is a Sequence Number that shows multiples of 13 written in 12 digit
strings.
All of the Sequence
Numbers used to produce counting sequences end in the digit one, so none of
them are divisible by 2 or 5. If you
want to find a Sequence Number to use with a number that is a multiple of 2 or
5 you will have to add zeros to the end of that Sequence Number. The number of zeros should be half the length
of the Sequence Number.
So 998,001,000 can be used
to show multiples of 2, 4, 5, 8, 10, 20, 25, 40 and 125, along with any combinations
of the multiples of 2, 3, 5, or 37 that divide 998,001,000 evenly.
It you have to add zeros,
please remember that the string length will be half the length of the counting
sequence number that you start with, then add that number of zeros to the
counting sequence number.
Power Sequences:
For Power Sequence the length of the
strings in the decimal expansion will be equal to the length of the Sequence
Number.
Example: The Sequence Number 999,995 will
show powers of five listed in six digit strings (beginning with 5^{0}).
1/999995 = 0.000000000005000025000125000615
…
Combinatoric (or Combinatorial) Functions:
C (n, m) or the number of ways to choose m
items from a group of items.
The Sequences Numbers representing C (n, m)
will consist of some number of nines equal in length to the length of the
strings you desire, and raised to the
m+1 power.
Example: The Sequence Number equal to 999^{6}
will produce the terms for C (n, 5) writing in three digit strings.
999^{6} = 994,014,980,014,994,001.
1/994014980014994001 =
0.
000 000 000 000 000 001 006 021 056 126 …
000 000 000 000 000 001 006 021 056 126 …
Fibonacci Like Sequences:
If you will recall:
The Sequence Number for the Fibonacci
Sequence and related sequences are composed of two parts. To produce n digit sequences each part has to
be written with n digits.
Example: The Sequence Number
999,998,999,999 produces the Fibonacci Sequence written in six digit strings.
The Sequence Number for the Tribonacci
Sequence and related sequences are composed of three parts. To produce n digit strings each part has to
be n digit parts.
Example: The Sequence Number
999,999,998,999,999.998,999,999,999 produces the Tribonacci Sequence written in
nine digit strings.
Sequence Numbers for Tetranacci like
sequences must have four parts, and the length of each part must be equal to
each other. The number of digits in each
part will be the number of digits in each string in the decimal expansion.
This pattern continues for other Fibonacci
like sequences (Pentanacci, Hexanacci, etc.)
David
1/999995 = 0.000001000005000025000125000625 …
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