Thursday, August 13, 2015

A Lovely Assortment of Sequence Numbers to Share with Your Students

8/13/2015

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 = 3062 - 1112 = 3622 - 2232 = 4062 - 2892 = 6582 - 5932 = 9262 - 8812 = 10622 - 10232 = 27182 - 27032 = 31342 - 31212 = 45222 - 45132 = 81342 - 81292 = 135542 - 135512 = 406582 - 406572.

81315 is a 8133-gonal and 27106-gonal 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 nine-hundred ninety-nine trillion, nine-hundred ninety-nine billion, nine-hundred ninety-nine million, nine-hundred and ninety-nine thousand, nine-hundred ninety-seven 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(n-1) + a(n-3) + 5 * a(n-4).  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 = 132), which produces a “13-nacci 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 5-Dimensional Hypercubes in an n-Dimensional 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 6-Dimensional 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 Number-ish 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 (12, 22, 32, …), 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 (13, 23, 33, …), 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 (14, 24, 34, …), 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 (15, 25, 35, …), 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 (16, 26, 36, …), 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 (17, 27, 37, …), 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 (18, 28, 38, …), written in nine digit strings.




David

16 comments:

  1. 142,857,142,856,857,142,857,143 produces a list of the multiples of seven, written in twelve digit strings.

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  2. 9,089,091 produces a list of the multiples of 11, written in 4 digit strings.

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  3. The 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.

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  4. The fraction 333333334666666667/333333332000000001999999998666666667 produces a digital expansion that shows the cube. , written in 9 digit strings.

    The fraction 333333333337000000000003666666666667/333333333331666666666669999999999996666666666668333333333333 produces a digital expansion that shows the 4th powers. , written in 12 digit strings.

    The fraction 333333333342000000000022000000000008666666666667/333333333331333333333338333333333326666666666671666666666664666666666667 produces a digital expansion that shows the 5th powers. , written in 12 digit strings.

    The fraction 111111111117444444444478000000000033555555555561888888888889/111111111110333333333335666666666662777777777781666666666664333333333334111111111111 produces a digital expansion that shows the 6th powers. , written in 12 digit strings.

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  5. The fraction 15873015874920634920653825396825435174603174622079365079366984126984127/15873015872888888888889333333333332444444444445555555555554666666666667111111111110984126984127 produces a digital expansion that shows the 7th powers. , written in 12 digit strings.

    The fraction 111111111111138555555555556032555555555557291000000000001735444444444444921444444444444471888888888888889/111111111111110111111111111115111111111111101777777777777791777777777777763777777777777787111111111111107111111111111112111111111111111 produces a digital expansion that shows the 8th powers. , written in 15 digit strings.

    The fraction 1763668430335097887125220458553817650793650793650949266313932980599922733686067019400508349206349206349232112874779541446208998236331569664903/1763668430335096984126984126984127063492063492063491851851851851851852222222222222222221777777777777777778148148148148148147936507936507936508015873015873015872998236331569664903 produces a digital expansion that shows the 9th powers. , written in 18 digit strings.

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  6. The fraction 12345679012345679024851851851851851852442469135802469135808088790123456790123472967333333333333333349510543209876543209882162864197530864197531454814814814814814814827320987654320987654321/12345679012345679012209876543209876543210555555555555555555553518518518518518518522592592592592592592586888888888888888888894592592592592592592588518518518518518518520555555555555555555554876543209876543209876679012345679012345679 produces a digital expansion that shows the 10th powers. , written in 21 digit strings.

    The fraction 12345679012345679037481481481481481483365888888888888888916092444444444444444564668074074074074074268200592592592592592712816222222222222222249425777777777777777779662185185185185185185210320987654320987654321/12345679012345679012197530864197530864198345679012345679012342962962962962962962969074074074074074074064296296296296296296307703703703703703703693925925925925925925932037037037037037037034320987654320987654321802469135802469135802320987654320987654321 produces a digital expansion that shows the 11th powers. , written in 21 digit strings.

    The fraction 53444497888942333386996045107156218267329404001443001443001443001987477152477152477152480696834696168029501362843381555555555555555555564240943081609748276414946625967291967291967291967836443001443001443001443027003954892843781732670621777724333279888835444391/53444497888942333386777136443803110469777136447971781305114638447971766019988242210464432686693121693121693121693121624338624338624338624338716049382716049382716049291005291005291005291005359788359788359788359788321575543797766019988242225749559082892416225749554914221580888247554914222275666720111164555609 produces a digital expansion that shows the 12th powers. , written in 24 digit strings.

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  7. 909,090,890,909,091 produces a list of the multiples of 11, written in 8 digit strings.

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  8. 3,448,275,862,068,965,517,241,379,309,655,172,413,793,103,448,275,862,069 produces a list of the multiples of 29 , written in 28 digit strings.

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  9. 2,439,024,389,756,097,561 produces a list of the multiples of 41 , written in 10 digit strings.

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  10. 2,127,659,574,468,085,106,382,978,723,404,255,319,148,936,169,787,234,042,553,191,489,361,702,127,659,574,468,085,106,383 produces a list of the multiples of 47 , written in 46 digit strings.

    20,408,163,265,306,122,448,979,591,836,734,693,877,550,979,591,836,734,693,877,551,020,408,163,265,306,122,449 produces a list of the multiples of 49 , written in 42 digit strings.

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  11. 1,694,915,254,237,288,135,593,220,338,983,050,847,457,627,118,644,067,796,609,830,508,474,576,271,186,440,677,966,101,694,915,254,237,288,135,593,220,339 produces a list of the multiples of 59 , written in 58 digit strings.


    1,639,344,262,295,081,967,213,114,754,098,360,655,737,704,918,032,786,885,245,898,360,655,737,704,918,0327,868,852,459,016,393,442,622,950,819,672,131,147,541 produces a list of the multiples of 61 , written in 60 digit strings.

    15,873,015,872,984,126,984,127 produces a list of the multiples of 63 , written in 12 digit strings.

    14,925,373,134,328,358,208,955,223,880,596,985,074,626,865,671,641,791,044,776,119,403 produces a list of the multiples of 67 , written in 33 digit strings.

    140,845,070,422,535,211,267,605,633,802,816,898,591,549,295,774,647,887,323,943,661,971,831 produces a list of the multiples of 71 , written in 35 digit strings.

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  12. 12,987,012,986,987,012,987,013 produces a list of the multiples of 77 , written in 12 digit strings.

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  13. 12,345,678,987,654,321 produces a list of the multiples of 81 , written in 9 digit strings.


    12,345,679,012,320,987,654,321 produces a list of the multiples of 81 , written in 12 digit strings.


    12,345,679,012,345,678,987,654,320,987,654,321 produces a list of the multiples of 81 , written in 18 digit strings.


    120,481,927,710,843,373,493,975,903,614,457,831,325,298,795,180,722,891,566,265,060,240,963,855,421,686,747 produces a list of the multiples of 83 , written in 41 digit strings.


    1,149,425,287,356,321,839,080,459,769,885,057,471,264,367,816,091,954,023 produces a list of the multiples of 87 , written in 28 digit strings.

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  14. 999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,998,999,999,999,999 shows a list of the Dodecanacci Sequence , written in 12 digit strings.


    999,999,999,998,999,999,999,997,999,999,999,996,999,999,999,995,999,999,999,994,999,999,999,993,999,999,999,992,999,999,999,991,999,999,999,990,999,999,999,989,999,999,999,988,999,999,999,988 shows a list of the Dodecanacci like sequence defined as: a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = a(6) = a(7) = a(8) = a(9) = a(10) = 0 , a(11) = 1 , and when n>11 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) , written in 12 digit strings.

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  15. 999,999,999,987,000,000,000,077,999,999,999,714,000,000,000,714,999,999,998,713,000,000,001,715,999,999,998,284,000,000,001,286,999,999,999,285,000,000,000,285,999,999,999,922,000,000,000,012,999,999,999,999 produces a list of the values of C(n,12) (the number of ways to select 12 items from a group of n items) , written in 12 digit strings.

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