8132015 is a composite, deficient, evil, happy, odd, polite, unprimeable, and wasteful number.
81315 can be written as the difference of two squares in 12 different ways: 81315 = 306^{2}  111^{2} = 362^{2}  223^{2} = 406^{2}  289^{2} = 658^{2}  593^{2} = 926^{2}  881^{2} = 1062^{2}  1023^{2} = 2718^{2}  2703^{2} = 3134^{2}  3121^{2} = 4522^{2}  4513^{2} = 8134^{2}  8129^{2} = 13554^{2}  13551^{2} = 40658^{2}  40657^{2}.
81315 is a 8133gonal and 27106gonal number.
Examples
of Sequence Numbers:
A sequence number is an
integer that produces recognizable sequence.
To produce this sequence you take the inverse of the sequence number,
and calculate the digital expansion. Example:
9801 is a sequence
number. The inverse of 9801 is
1/9801. When you change this fraction
into a decimal number you get 0.0001020304050607… This sequence counts, using 2 digit
strings, from 00 to 97, without error.
If you would like to
check some of these out I invite you to do so. The mathematics is simple. You don’t have to do anything harder than
adding, subtracting, multiplying and dividing. But for many of them you will want to do it
on your computer – I would recommend using the Wolfram Alpha website: www.wolframalpha.com .
I also urge you to use
the Online Encyclopedia of Integer Sequences website to verify that the digit
sequence you get matches what the sequence is supposed to be giving: www.oeis.org .
The following is a list
of some of the known sequence number, and the digital sequences they produce.

SEQUENCE
NUMBERS THAT PRODUCE COUNTING SEQUENCES:
81 produces a digital sequence that counts
from 0 to 7, written in one digit strings.
9,801 produces a digital sequence that
counts from 00 to 97, written in two digit strings.
998,001 produces a digital sequence that
counts from 000 to 997, written in three digit strings.
99,980,001 produces a digital sequence that
counts from 0000 to 9997, written in four digit strings.
9,999,800,001 produces a digital sequence
that counts from 00000 to 99997, written in five digit strings.
This patterns continues.
999,999,999,999,998,000,000,000,000,001
produces a digital sequence that counts from 000000000000 to 999999999997
without error, written in 12 digit strings.
It does it through one division operation: 1/999999999999998000000000000001. Counting from zero to ninehundred
ninetynine trillion, ninehundred ninetynine billion, ninehundred
ninetynine million, ninehundred and ninetynine thousand, ninehundred
ninetyseven without error! I don’t care
if there is a mathematical explanation for how this is possible, it is still
pretty AMAZING.
But, hold on. There are more!
SEQUENCE
NUMBER THAT SHOW MULTIPLES OF N:
333,332,666,667 produces a list of the
multiples of three, written in six digit strings.142,856,857,143 produces a
list of the multiples of seven, written in six digit strings.
90,908,909,091 produces a list of the
multiples of 11, written in six digit strings.
76,922,923,077 produces a list of the
multiples of 13, written in six digit strings.
5,882,352,941,176,469,411,764,705,882,353
produces a list of the multiples of 17, written in 16 digit strings.
52,631,578,947,368,420,947,368,421,052,631,579
produces a list of the multiples of 19, written in 18 digit strings.
47,618,952,381 produces a list of the
multiples of 21, written in six digit strings.
4,347,826,086,956,521,739,129,565,217,391,304,347,826,087
produces a list of the multiples of 23, written in 22 digit strings.
37,036,962,963 produces a list of the
multiples of 27, written in six digit strings.
32,258,064,516,128,967,741,935,483,871
produces a list of the multiples of 31, written in 15 digit strings.
30,302,969,697 produces a list of the
multiples of 33, written in six digit strings.
27,026,972,973 produces a list of the
multiples of 37, written in six digit strings.
25,640,974,359 produces a list of the
multiples of 39, written in six digit strings.
243,897,561 produces a list of the
multiples of 41, written in five digit strings.
23,255,813,953,488,372,092,976,744,186,046,511,627,907
produces a list of the multiples of 43, listed in 21 digit strings.
20,408,122,449 produces a list of the multiples
of 49, written in six digit strings.
1,960,784,313,725,489,803,921,568,627,451
produces a list of the multiples of 51, written in 16 digit strings.
1,886,792,452,829,811,320,754,717 produces
a list of the multiples of 53, written in 13 digit strings.
17,543,859,649,122,806,982,456,140,350,877,193
produces a list of the multiples of 57, written in 18 digit strings.
15,872,984,127 produces a list of the
multiples of 63, written in six digit strings.
1,449,275,362,318,840,579,709,855,072,463,768,115,942,029
produces a list of the multiples of 69, written in 22 digit strings.
136,986,298,630,137 produces a list of the
multiples of 73, written in eight digit strings.
12,986,987,013 produces a list of the
multiples of 77, written in six digit strings.
1,265,822,784,809,873,417,721,519 produces
a list of the multiples of 79, written in 13 digit strings.
12,345,654,321 produces a list of the
multiples of 81, written in six digit strings.
10,989,010,988,989,010,989,011 produces a
list of the multiples of 91, written in 12 digit strings.
10,752,688,172,042,989,247,311,827,957
produces a list of the multiples of 93, written in 15 digit strings
10,100,989,899 produces a list of the
multiples of 99, written in six digit strings.
99,009,899,009,901 produces a list of the
multiples of 101, written in eight digit strings.
9,008,990,991 produces a list of the
multiples of 111, written in six digit strings.

SEQUENCE
NUMBERS THAT SHOW POWERS OF NUMBERS:
999,999,999,998 produces a list of the
powers of two, written in 12 digit strings.
999,999,997 produces a list of the powers
of three, written in nine digit strings.
999,999,993 produces a list of the powers
of seven, written in nine digit strings.
9,999,999,999,987 produces a list of the
powers of 13, written in 13 digit strings.
39,999,999,999,999 produces a list of the
powers of 25, written in 15 digit strings.
999,999,999,999,999,877 produces a list of
the powers of 123, written in 18 digit strings.
SEQUENCE
NUMBERS THAT SHOW COMBINATORIAL FUNCTIONS:
999,999,995,000,000,009,999,999,990,000,000,004,999,999,999
produce the values of C (n, 4) (the number of ways of selecting 4 items from a
group of n items), written in nine digit strings.
999,999,994,000,000,014,999,999,980,000,000,014,999,999,994,
000,000,001 produces a list of the values of C (n, 5) (the number of ways of
selecting 5 items from a group of n items, written in nine digit strings.
999,999,910,000,003,599,999,916,000,001,259,999,987,400,000,083,999, 999,640,000,000,899,999,999
produces a list of the values of C (n, 8) (the number of ways to select 8 items
from a group of n items), written in eight digit strings.
99,999,999,999,860,000,000,000,090,999,999,999,963,600,000,
000,010,009,999,999,997,998,000,000,000,300,299,999,999,965, 680,000,000,003,002,999,999,999,799,800,000,000,010,009,999,
999,999,636,000,000,000,009,099,999,999,999,860,000,000,000, 001 produces a
list of the values of C (n, 13) (the number of ways to select 13 items from a
group of n items, written in 13 digit sequences.
SEQUENCE
NUMBERS THAT SHOW FIBONACCI AND FIBONACCI LIKE SEQUENCES:
999,999,998,999,999,998,999,999,999 shows a
list of the Tribonacci numbers (using the Tribonacci number sequence that
starts with 0, 1, and 1), written in nine digit strings.
999,997,999,999 shows a list of the numbers
in the Pell Sequence, written in six digit strings.
999,998,999,998 shows a list of the numbers
in the Jacobsthal Sequence, written in six digit strings.
999,999,998,999,999,998,999,999,999 shows a
list of the Tribonacci numbers (using the Tribonacci number sequence that
starts with 0, 1, and 1), written in nine digit strings.
999,999,997,999,999,999,998,999,999,994
produces the 2,0,1,5 Tetranacci sequence, a Tetranacci like sequence defined
as: a(0) = a(1) = a(2) = 0, a(3) = 1, and when n>3 then a(n) = 2 * a(n1) +
a(n3) + 5 * a(n4). It’s terms are
written in nine digit strings.
9,999,999,999,998,999,999,999,999,899,999,999,999,989,999,999,
999,998,999,999,999,999,899,999,999,999,989,999,999,999,998, 999,999,999,999,899,999,999,999,989,999,999,999,998,999,999,
999,999,899,999,999,999,989,999,999,999,999 shows a list of the Tridecanacci
numbers (from the Tridecanacci sequence that starts with 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, and 1), written in 13 digit strings.
9,999,999,999,998,999,999,999,999,799,999,999,999,969,999,999,
999,995,999,999,999,999,499,999,999,999,939,999,999,999,992, 999,999,999,999,199,999,999,999,909,999,999,999,989,999,999,
999,998,899,999,999,999,879,999,999,999,987 shows a list of the Tridecanacci
like sequence defined as: a(0) = a(1) = a(2) = a(3) = … = a(11) = 0, a(12) = 1,
and when n>12 then a(n) = a(n – 1) + 2*a(n – 2) + 3*a(n – 3) + 4*a(n – 4) +
5*a(n – 5) + 6*a(n – 6) + 7*a(n – 7) + 8*a(n – 8) + 9*a(n – 9) + 10*a(n – 10) +
11*a(n – 11) + 12*a(n – 12) + 13*a(n – 13).
Each term is written as a 13 digit string.
WOW, a 169 digit sequence number (169 = 13^{2}),
which produces a “13nacci sequence”, where each term is the sum of the 13
previous terms, each of the 13 previous terms are multiplied by a different
number (from 1 to 13), and written in 13 digit strings. Triskaidekaphiliacs Rule!
MISCELANIOUS
SEQUENCE FUNCTIONS:
999,999,997,000,000,002,999,999,999
produces a list of the triangular numbers, written in nine digit strings.
999,999,996,000,000,005,999,999,996,000,000,001
produces a list of the tetrahedral numbers, written in nine digit strings.
999,999,988,000,000,059,999,999,840,000,000,239,999,999,
808,000,000,064 produces a digit sequence that shows the number of 5Dimensional
Hypercubes in an nDimensional Hypercube, written in 9 digit strings.
999,999,986,000,000,083,999,999,720,000,000,559,999,999,328,
000,000,447,999,999,872 produces a list of the number of 6Dimensional Hypercubes
in an (n + 6) Dimensional Hypercube.
SEQUENCES
THAT RESULT FROM MULTIPLE SEQUENCE NUMBERS OR IRREDUCIBLE FRACTIONS:
Two special cases that do not fit my
definition of a Sequence Number are below.
Both show Sequence Numberish properties.
4,999,990,000,005
produces a list of the multiples of two, written in six digit strings, but
will have an extra zero at the beginning.
Possibly a better way of producing the multiples of two is by adding
the results produced by 499,999,000,001, and 499,998,000,003,499,997,000,001. This will also show the multiples of two,
written in six digit strings, but does not produce an extra the extra zero at
the beginning.
1,999,996,000,002 will
produce a list of the multiples of five, written in six digit strings, but
adds an extra zero at the beginning.
Perhaps a better way of producing multiples of five is by adding the
results produced by 199,999,600,001, 49,999,800,000,499,999,400,001, and
3,333,306,666,786,666,320,000,736,665,533,334,646,665,666,667. It also shows the multiples of five,
written in six digit strings, but does not need an extra zero at the
beginning.

Adding the results
produced by 500,000, 999,997,000,005, and
99,999,600,000,699,999,799,999,500,000 will show the numbers in the Lucas
Sequence, written in six digit strings.

Adding the results
produced by 999,996,000,007, 124,999,125,002,749,995,750,003,125,000, and 17,856,892,858,803,564,714,303,749,966,910,755,446,395,464,
301,124,996,875,000 will show sequence of the squares (1^{2}, 2^{2},
3^{2}, …), written in six digit strings.

The following fractions
are irreducible, and do not lend themselves easily for conversion to multiple
sequence numbers, but they do produce interesting results.
The fraction 333,346,666,700,000
/ 3,332,000,019,999,866,667 produces a digital expansion that shows the cubes
(1^{3}, 2^{3}, 3^{3}, …), written in five digit
strings.
The fraction 33,337,000,036,666,700,000
/ 3,333,166,669,999,966,666,833,333 produces a digital expansion that shows
the fourth powers (1^{4}, 2^{4}, 3^{4}, …), written
in five digit strings.
The fraction 333,342,000,022,000,008,666,667,000,000
/ 333,331,333,338,333,326,666,671,666,664,666,667 produces a digital
expansion that shows the fifth powers (1^{5}, 2^{5}, 3^{5},
…), written in six digit strings.
The fraction 111,111,744,444,780,000,033,555,556,188,888,890,000,000
/ 1,111,110,333,333,566,666,627,777,781,666,666,433,333,341,
111,111 produces a digital expansion that shows the sixth powers (1^{6}, 2^{6}, 3^{6}, …), written in seven digit strings.
The fraction 11,111,124,444,445,767,777,804,622,222,354,555,555,688,888,
888,900,000,000 / 1,111,111,022,222,225,333,333,271,111,111,888,888,882,666, 666,697,777,777,688,888,889 produces a digital expansion that shows the seventh powers (1^{7}, 2^{7}, 3^{7}, …), written in eight digit strings.
The fraction 111,111,138,555,556,032,555,557,291,000,001,735,444,444,921,
444,444,471,888,888,889,000,000,000 / 111,111,110,111,111,115,111,111,101,777,777,791,777,777,763,
777,777,787,111,111,107,111,111,112,111,111,111 produces a digital expansion
that shows the eighth powers (1^{8}, 2^{8}, 3^{8}, …),
written in nine digit strings.

David
142,857,142,856,857,142,857,143 produces a list of the multiples of seven, written in twelve digit strings.
ReplyDelete9,089,091 produces a list of the multiples of 11, written in 4 digit strings.
ReplyDeleteThe fraction 333,346,666,700,000 / 33,332,000,019,999,866,667 produces a digital expansion that shows the cubes (13, 23, 33, …), written in five digit strings.
ReplyDelete1
ReplyDelete