Generating
Counting Sequences Using Sequence Numbers
David Brooks
SAWB, BE, MS, MS
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.
|
92 = 81
81 is a Sequence Number that produces a
counting sequence, that counts from zero to seven, in single digit strings,
before making an error (it skips “8”).
It forms a repeating decimal with a period of nine digits.
1/81 =
0.
0 1 2 3 4 5 6 7 9 0 1 2 3 …
(Spaces have been added to separate the
terms of the sequence and make them easier to read.)
|
|
992 = 9,801
9,801 is a Sequence
Number that counts from zero to 97, using two digit strings, before making an
error (it skips “98”). It forms a
repeating decimal with a period of 198 digits.
1/9801 =
0.
00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 99 00 01 02 03 … |
|
9992 = 998,001
998,001 is a Sequence
Number that produces a counting sequence from zero to 997, using three digit
strings, before making an error. It
forms a repeating decimal with a period of 2,997 digits.
1/998001 =
0.
000 001 002 003 004 005 006 007 008 009 010 011 012 013 014 015 016 017 018 019 020 021 022 023 024 025 026 027 028 029 030 031 032 033 034 035 036 037 038 039 040 041 042 043 044 045 046 047 048 049 050 051 052 053 054 055 056 057 058 059 060 061 062 063 064 065 066 067 068 069 070 071 072 073 074 075 076 077 078 079 080 081 082 083 084 085 086 087 088 089 090 091 092 093 094 095 096 097 098 099 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 999 000 001 002 003 … |
|
9,9992 =
99,980,001
99,999,001 is a Sequence
Number that produces a counting sequence, counting from zero to 9,997, using
four digit strings, before it makes an error (it skips 9998). It produces a repeating decimal with a
period of 39,996 digits.
1/99980001 =
0.
0000 0001 0002 0003 0004 0005 0006 0007 0008 0009 0010 0011 0012 0013 0014 0015 0016 0017 0018 0019 0020 0021 0022 0023 0024 0025 0026 0027 0028 0029 0030 0031 0032 0033 0034 0035 0036 0037 0038 0039 0040 0041 0042 0043 0044 0045 0046 0047 0048 0049 0050 0051 0052 0053 0054 0055 0056 0057 0058 0059 0060 0061 0062 0063 0064 0065 0066 0067 0068 0069 0070 0071 0072 0073 0074 0075 0076 0077 0078 0079 0080 0081 0082 0083 0084 0085 0086 0087 0088 0089 0090 0091 0092 0093 0094 0095 0096 0097 0098 0099 0100 0101 0102 0103 0104 0105 0106 0107 0108 0109 0110 0111 0112 0113 ...
Well, you get the idea …
|
David
Appendix:
Sequence Numbers that generate counting
sequences:
|
Sequence
Number
|
String
Length
|
|
99,980,001
|
Produces a counting Sequence written
with 4 digit strings.
|
|
9,999,800,001
|
Produces a counting Sequence written
with 5 digit strings.
|
|
999,998,000,001
|
Produces a counting Sequence written
with 6 digit strings.
|
|
99,999,980,000,001
|
Produces a counting Sequence written
with 7 digit strings.
|
|
9,999,999,800,000,001
|
Produces a counting Sequence written
with 8 digit strings.
|
|
999,999,998,000,000,001
|
Produces a counting Sequence written
with 9 digit strings.
|
|
99,999,999,980,000,000,001
|
Produces a counting Sequence written
with 10 digit strings.
|
|
9,999,999,999,800,000,000,001
|
Produces a counting Sequence written
with 11 digit strings.
|
|
Sequence
Number
|
String
Length
|
|
999,999,999,998,000,000,000,
001 |
Produces a counting Sequence written
with 12 digit strings.
Warning: You are now exceeding one
hundred billion (the estimated number of galaxies in our observable
universe).
|
|
99,999,999,999,980,000,000,
000,001 |
Produces a counting Sequence written
with 13 digit strings.
|
|
9,999,999,999,999,800,000,000,000,001
|
Produces a counting Sequence written
with 14 digit strings.
|
|
999,999,999,999,998,000,000,
000,000,001 |
Produces a counting Sequence written
with 15 digit strings.
|
|
99,999,999,999,999,980,000,
000,000,000,001 |
Produces a counting Sequence written
with 16 digit strings.
|
|
9,999,999,999,999,999,800,000,000,000,000,001
|
Produces a counting Sequence written
with 17 digit strings.
|
|
999,999,999,999,999,998,000,
000,000,000,000,001 |
Produces a counting Sequence written
with 18 digit strings.
|
|
99,999,999,999,999,999,980,
000,000,000,000,000,001 |
Produces a counting Sequence written
with 19 digit strings.
|
|
9,999,999,999,999,999,999,800,000,000,000,000,000,001
|
Produces a counting Sequence written
with 20 digit strings.
|
|
Sequence
Number
|
String
Length
|
|
999,999,999,999,999,999,998,
000,000,000,000,000,000,001 |
Produces a counting Sequence written
with 21 digit strings.
|
|
99,999,999,999,999,999,999,
980,000,000,000,000,000,000, 001 |
Produces a counting Sequence written
with 22 digit strings.
Warning: You are now exceeding one
thousand million million million (the estimated number of stars in our
observable universe).
|
|
9,999,999,999,999,999,999,999,800,000,000,000,000,000,000,
001 |
Produces a counting Sequence written
with 23 digit strings.
|
|
999,999,999,999,999,999,999,
998,000,000,000,000,000,000, 000,001 |
Produces a counting Sequence written
with 24 digit strings.
|
|
99,999,999,999,999,999,999,
999,980,000,000,000,000,000, 000,000,001 |
Produces a counting Sequence written
with 25 digit strings.
Warning: You are now exceeding the
ability to count past one million million million million.
|
|
9,999,999,999,999,999,999,999,999,800,000,000,000,000,000,
000,000,001 |
Produces a counting Sequence written
with 26 digit strings.
|
|
Sequence
Number
|
String
Length
|
|
999,999,999,999,999,999,999,999,998,000,000,000,000,000,000,000,000,001
|
Produces a counting Sequence written
with 27 digit strings.
|
|
99,999,999,999,999,999,999,999,999,980,000,000,000,000,000,000,000,000,001
|
Produces a counting Sequence written
with 28 digit strings.
|
|
9,999,999,999,999,999,999,999,999,999,800,000,000,000,000,000,000,000,000,001
|
Produces a counting Sequence written
with 29 digit strings.
|
|
999,999,999,999,999,999,999,
999,999,998,000,000,000,000, 000,000,000,000,000,001 |
Produces a counting Sequence written
with 30 digit strings.
|
|
99,999,999,999,999,999,999,
999,999,999,980,000,000,000, 000,000,000,000,000,000,001 |
Produces a counting Sequence written
with 31 digit strings.
Warning: You are now exceeding the
ability to count past one million million million million million.
|
|
99,999,999,99,999,999,999,999,999,999,999,800,000,000,000,
000,000,000,000,000,000,001 |
Produces a counting Sequence written
with 32 digit strings.
|
|
Sequence
Number
|
String
Length
|
|
999,999,999,999,999,999,999,
999,999,999,998,000,000,000, 000,000,000,000,000,000,000, 001 |
Produces a counting Sequence written
with 33 digit strings.
|
|
99,999,999,999,999,999,999,
999,999,999,999,980,000,000, 000,000,000,000,000,000,000, 000,001 |
Produces a counting Sequence written
with 34 digit strings.
|
|
9,999,999,999,999,999,999,999,999,999,999,999,800,000,000,
000,000,000,000,000,000,000, 000,001 |
Produces a counting Sequence written
with 35 digit strings.
|
|
999,999,999,999,999,999,999,
999,999,999,999,998,000,000, 000,000,000,000,000,000,000, 000,000,001 |
Produces a counting Sequence written
with 36 digit strings.
|
|
99,999,999,999,999,999,999,
999,999,999,999,999,980,000, 000,000,000,000,000,000,000, 000,000,000,001 |
Produces a counting Sequence written
with 37 digit strings.
Warning: You are now exceeding the
ability to count past one million million million million, million million.
|
|
9,999,999,999,999,999,999,999,999,999,999,999,999,800,000,
000,000,000,000,000,000,000, 000,000,000,001 |
Produces a counting Sequence written
with 38 digit strings.
|
|
Sequence
Number
|
String
Length
|
|
999,999,999,999,999,999,999,
999,999,999,999,999,998,000, 000,000,000,000,000,000,000, 000,000,000,000,001 |
Produces a counting Sequence written
with 39 digit strings.
|
|
99,999,999,999,999,999,999,
999,999,999,999,999,999,980, 000,000,000,000,000,000,000, 000,000,000,000,000,001 |
Produces a counting Sequence written
with 40 digit strings.
|
|
9,999,999,999,999,999,999,999,999,999,999,999,999,999,800,
000,000,000,000,000,000,000, 000,000,000,000,000,001 |
Produces a counting Sequence written
with 41 digit strings.
|
|
999,999,999,999,999,999,999,
999,999,999,999,999,999,998, 000,000,000,000,000,000,000, 000,000,000,000,000,000,001 |
Produces a counting Sequence written
with 42 digit strings.
|
|
99,999,999,999,999,999,999,
999,999,999,999,999,999,999, 980,000,000,000,000,000,000, 000,000,000,000,000,000,000, 001 |
Produces a counting Sequence written
with 43 digit strings.
Warning: You are now exceeding the
ability to count past one million million million million, million million
million.
|
|
Sequence
Number
|
String
Length
|
|
9,999,999,999,999,999,999,999,999,999,999,999,999,999,999,
800,000,000,000,000,000,000, 000,000,000,000,000,000,000, 001 |
Produces a counting Sequence written
with 44 digit strings.
|
|
999,999,999,999,999,999,999,
999,999,999,999,999,999,999, 998,000,000,000,000,000,000, 000,000,000,000,000,000,000, 000,001 |
Produces a counting Sequence written
with 45 digit strings.
|
|
99,999,999,999,999,999,999,
999,999,999,999,999,999,999, 999,980,000,000,000,000,000, 000,000,000,000,000,000,000, 000,000,001 |
Produces a counting Sequence written
with 46 digit strings.
Warning: You are now exceeding the
ability to count past one billion billion billion billion billion.
|
|
9,999,999,999,999,999,999,999,999,999,999,999,999,999,999,
999,800,000,000,000,000,000, 000,000,000,000,000,000,000, 000,000,001 |
Produces a counting Sequence written
with 47 digit strings.
|
|
999,999,999,999,999,999,999,
999,999,999,999,999,999,999, 999,998,000,000,000,000,000, 000,000,000,000,000,000,000, 000,000,000,001 |
Produces a counting Sequence written
with 48 digit strings.
|
|
Sequence
Number
|
String
Length
|
|
99,999,999,999,999,999,999,
999,999,999,999,999,999,999, 999,999,980,000,000,000,000, 000,000,000,000,000,000,000, 000,000,000,000,001 |
Produces a counting Sequence written
with 49 digit strings.
Warning: You are now exceeding the
ability to count past one million million million million million million
million million.
|
|
9,999,999,999,999,999,999,999,999,999,999,999,999,999,999,
999,999,800,000,000,000,000, 000,000,000,000,000,000,000, 000,000,000,000,001 |
Produces a counting Sequence written
with 50 digit strings.
|
1/9999800001 is a Sequence Number that counts from zero to 99997
ReplyDeleterepeating decimal with a period of 499,995 digits.
1/9999800001
ReplyDeletecounts from 00000 TO 99997
http://bill16888.blogspot.hk/2017/06/19999800001.html
1/9999800001 =
ReplyDelete0 . 00000 00001 00002 00003 00004 00005 00006 00007 00008 00009 00010 00011 00012 00013 00014 00015 00016 00017 00018 00019 00020 00021 00022 00023 00024 00025 00026 00027 00028 00029 00030 00031 00032 00033 00034 00035 00036 00037 00038 00039 00040 00041 00042 00043 00044 00045 00046 00047 00048 00049 00050 ...