Sequence Numbers that Show Multiples of a Number

9/12/2011

SEQUENCE NUMBERS THAT SHOW MULTIPLES OF A NUMBER:

(Can sequence numbers do more than count?)

Can we adapt what we know about counting sequences to create sequences that show multiples of numbers (other than one)?

Suppose we want to have a digital sequence that shows multiples of 3 (or as some people would say “counts by threes”).  I could start with the sequence number 998001, and its inverse 1/998001.  I haven’t changed anything yet – so I know it still counts by ones.

But if I do change it by multiplying the fraction by 3 (3/998001) well, this is not the inverse of an integer – we need to have a 1 on top of the fraction.  That’s OK here, because 3/998001 simplifies (the numerator and the denominator are both divisible by 3) and I get 1/332667.

Multiples of Three The sequence number is 332,667. 1/332667 = 0. 000  003  006  009  012  015  018  021  024  027  030  033  036  039  042  045  048  051  054  057  060  063  066  069  072  075  078  081  084  087  090  093  096  099  102  105  108  111  114  117  120  123  126  129  132  135  138  141  144  147  150  153  156  159  162  165  168  171  174  177  180  183  186  189  192  195  198  201  204  207  210  213  216  219  222  225  228  231  234  237  240  243  246  249  252  255  258  261  264  267  270  273  276  279  282  285  288  291  294  297  300  303  306  309  312  315  318  321  324  327  330  333  336  339  342  345  348  351  354  357  360  363  366  369  372  375  378  381  384  387  390  393  396  399  402  405  408  411  414  417  420  423  426  429  432  435  438  441  444  447  450  453  456  459  462  465  468  471  474  477  480  483  486  489  492  495  498  501  504  507  510  513  516  519  522  525  528  531  534  537  540  543  546  549  552  555  558  561  564  567  570  573  576  579  582  585  588  591  594  597  600  603  606  609  612  615  618  621  624  627  630  633  636  639  642  645  648  651  654  657  660  663  666  669  672  675  678  681  684  687  690  693  696  699  702  705  708  711  714  717  720  723  726  729  732  735  738  741  744  747  750  753  756  759  762  765  768  771  774  777  780  783  786  789  792  795  798  801  804  807  810  813  816  819  822  825  828  831  834  837  840  843  846  849  852  855  858  861  864  867  870  873  876  879  882  885  888  891  894  897  900  903  906  909  912  915  918  921  924  927  930  933  936  939  942  945  948  951  954  957  960  963  966  969  972  975  978  981  984  987  990  993  … Written in three digit strings. This is a repeating decimal with a period of 999. Accurate to the 331st non-zero term (331 * 3 = 993). Compare with OEIS sequence A008585.

We were counting by ones, and we multiplied by three so now we are counting by threes (or multiples of three).  And the 3 in the numerator conveniently canceled with a factor of three in the denominator.  So we are good to go with this example.

So the secret here is to know which numbers will factor out and which won’t.  Threes will almost always factor out (unless you have to many threes to factor out).  Twos and fives will never factor out (the sequence numbers we have use all end in with a 1 – so they can’t be divisible by 2 or by 5).

Let’s look as some of the other factors first:

9,801	34 * 112
998,001	36 * 372
99,980,001	34 * 112 * 1012
9,999,800,001	34 * 412 * 2712
999,998,000,001	36 * 72 * 112 * 132 * 372
99,999,980,000,001	34 * 2392 * 46492
9,999,999,800,000,001	3^4 * 11^2 * 732 * 1012 * 1372
999,999,998,000,000,001	38 * 372 * 333,6672
99,999,999,980,000,000,001	32 * 112 * 412 * 2712 * 90912
9,999,999,999,800,000,000,001	34 * 216492 * 513,2392
999,999,999,998,000,000,000, 001	3^6 * 72 * 112 * 132 * 372 * 1012 * 99012
99,999,999,999,980,000,000, 000,001	34 * 532 * 792 * 265,371,6532
9,999,999,999,999,800,000, 000,000,001	34 * 112 * 2392 * 46492 * 909,0912
999,999,999,999,998,000,000, 000,000,001	36 * 312 * 372 * 412 * 2712 * 2,906,1612
99,999,999,999,999,980,000, 000,000,000,001	34 * 112 * 172 * 732 * 1012 * 1372 * 5,882,3532
9,999,999,999,999,999,800,000, 000,000,000,001	34 * 2,071,7232 * 5,363,222,3572
999,999,999,999,999,998,000, 000,000,000,000,001	38 * 72 * 112 * 132 * 192 * 372 * 525792 * 333,6672
99,999,999,999,999,999, 980,000,000,000,000,000,001	34 * 1,111,111,111,111,111,1112
9,999,999,999,999,999, 999,800,000,000,000,000, 000,001	34 * 112 * 412 * 1012 * 2712 * 35412 * 90912 * 279612
999,999,999,999,999,999,998, 000,000,000,000,000,000,001	36 * 372 * 432 * 2392 * 19332 * 46492 * 10,838,6892
99,999,999,999,999,999,999, 980,000,000,000,000,000, 000,001	34 * 114 * 232 * 40932 * 87792 * 216492 * 513,2392
9,999,999,999,999,999,999, 999,800,000,000,000,000,000, 000,001	34 * 11,111,111,111,111,111,111, 1112
999,999,999,999,999,999,999, 998,000,000,000,000,000,000, 000,001	36 * 72 * 112 * 132 * 372 * 732 * 1012 * 1372 * 99012 * 99,990,0012
99,999,999,999,999,999,999, 999,980,000,000,000,000,000, 000,000,001	34 * 412 * 2712 * 214012 * 256012 * 182,521,213,0012
9,999,999,999,999,999,999, 999,999,800,000,000,000, 000,000,000,000,001	34 * 112 * 532 * 792 * 8592 * 2653716532 * 10583130492
999,999,999,999,999,999, 999,999,998,000,000,000, 000,000,000,000,000,001	310 * 372 * 7572 * 3336672 * 4403346547776312
99,999,999,999,999,999,999, 999,999,980,000,000,000,000, 000,000,000,000,001	34 * 112 * 292 * 1012 * 2392 * 2812 * 46492 * 9090912 * 1214994492
9,999,999,999,999,999,999,999, 999,999,800,000,000,000,000, 000,000,000,000,001	34 * 31912 * 167632 * 430372 * 620032 * 778438393972
999,999,999,999,999,999,999, 999,999,998,000,000,000,000, 000,000,000,000,000,001	36 * 72 * 112 * 132 * 312 * 372 * 412 * 2112 * 2412 * 2712 * 21612 * 90912 * 29061612

Notice that they don’t all have the same factors, so not every sequence number will work every time.  But if you look hard you may find some that do work for your specific situation.

999,998,000,001 has factors of 3, 7, 11, 13, and 37, so I can use it to find a sequence number that lists its term in 6 digit strings for multiples of 3, 7, 11, 13, and 37.  So 3/999998000001, 7/999998000001, 11/999998000001, 13/999998000001, and 37/999998000001 can all be used – they will all simplify.

But it will also work for combinations of these numbers 3 * 7 = 21, 3 * 11 = 33, 3 * 13 = 39, and 3 * 37 = 111.  7 * 11 = 77, 7 * 13 = 91, 7 * 37 = 259, 11 * 13 = 143, 11 * 37 = 407, and 13 * 37 = 481,  And don’t forget, all of these factors occur twice, except for 3 which occurs 6 times – so we can use cancel out numbers like 9, 42, 49, 99, 121, 169 and several others.

I’m not going to list all of the possibilities, but I will give you some examples.

Suppose I wanted a sequence number that would produce a sequence of terms that were all of the multiples of 123.  The prime factors of 123 are 3 and 41.  So I need to find a number in the table above that has factors of 3 and 41.  9,999,800,001 will work, but so will 99,999,999,980,000,000,001 and 999,999,999,999,998,000,000,000,000,001.  The first will produce terms written in five digit strings, the second in 10 digit strings, and the third in 15 digit strings.  Since multiples of 123 will grow faster that multiples of 1, I need to pick a string size that is appropriate (If you are not sure just guess, check, and try again until you find what you need.).  I’m going to try the middle one.  If I don’t like the results, I can change it

123/99999999980000000001 simplifies to 1/813008129918699187. So 813,008,129,918,699,187 will be our sequence number. 1/813008129918699187 = 0. 0000000000  0000000123  0000000246  0000000369  0000000492  0000000615  0000000738  0000000861  0000000984  0000001107  0000001230  0000001353  0000001476  0000001599  0000001722  0000001845  0000001968  0000002091  0000002214  0000002337  0000002460  0000002583  0000002706  0000002829  0000002952  0000003075  0000003198  0000003321  0000003444  0000003567  0000003690  0000003813  0000003936  0000004059  0000004182  0000004305  0000004428  0000004551  0000004674  0000004797  0000004920  0000005043  0000005166  0000005289  0000005412  0000005535  0000005658  0000005781  0000005904  0000006027  0000006150  0000006273  0000006396  0000006519  0000006642  0000006765  0000006888  0000007011  0000007134  0000007257  0000007380  0000007503  0000007626  0000007749  0000007872  0000007995  0000008118  0000008241  0000008364  0000008487  0000008610  0000008733  0000008856  0000008979  0000009102  0000009225  0000009348  0000009471  0000009594  0000009717  0000009840  0000009963  0000010086  0000010209  0000010332  0000010455  0000010578  0000010701  0000010824  0000010947  0000011070  0000011193  0000011316  0000011439  0000011562  0000011685  0000011808  0000011931  0000012054  0000012177  0000012300  0000012423  0000012546  0000012669  0000012792  0000012915  0000013038  0000013161  0000013284  0000013407  0000013530  0000013653  0000013776  0000013899  0000014022  0000014145  0000014268  0000014391  0000014514  0000014637  0000014760  0000014883  0000015006  0000015129  0000015252  0000015375  0000015498  0000015621  0000015744  0000015867  0000015990  0000016113  0000016236  0000016359  0000016482  0000016605  0000016728  0000016851  0000016974  0000017097  0000017220  0000017343  0000017466  0000017589  0000017712  0000017835  0000017958  0000018081  0000018204  0000018327  0000018450  0000018573  0000018696  0000018819  0000018942  0000019065  0000019188  0000019311  0000019434  0000019557  0000019680  0000019803  0000019926  0000020049  0000020172  0000020295  0000020418  0000020541  0000020664  0000020787  0000020910  0000021033  0000021156  0000021279  0000021402 0000021525  0000021648  0000021771  0000021894  0000022017  0000022140  0000022263  0000022386  0000022509  0000022632  0000022755  0000022878  0000023001  0000023124  0000023247  0000023370  0000023493  0000023616  0000023739  0000023862  0000023985  0000024108  0000024231  0000024354  0000024477  0000024600  0000024723  0000024846  0000024969  0000025092  0000025215  0000025338  0000025461  0000025584  0000025707  0000025830  0000025953  0000026076  0000026199  0000026322  0000026445  0000026568  0000026691  0000026814  0000026937  0000027060  0000027183  0000027306  0000027429  0000027552  0000027675  0000027798  0000027921  0000028044  0000028167  0000028290  0000028413  0000028536  0000028659  0000028782  0000028905  0000029028  0000029151  0000029274  0000029397  0000029520  0000029643  0000029766  0000029889  0000030012  0000030135  0000030258  0000030381  0000030504  0000030627  0000030750  0000030873  0000030996  0000031119  0000031242  0000031365  0000031488  0000031611  0000031734  0000031857  0000031980  0000032103  0000032226  0000032349  0000032472  0000032595  0000032718  0000032841  0000032964  0000033087  0000033210  0000033333  0000033456  0000033579  0000033702  0000033825  0000033948  0000034071  0000034194  0000034317  0000034440  0000034563  0000034686  0000034809  0000034932  0000035055  0000035178  0000035301  0000035424  0000035547  0000035670  0000035793  0000035916  0000036039  0000036162  0000036285  0000036408  0000036531  0000036654  0000036777  0000036900  0000037023  0000037146  0000037269  0000037392  0000037515  0000037638  0000037761  0000037884  0000038007  0000038130  0000038253  0000038376  0000038499  0000038622  0000038745  0000038868  0000038991  0000039114  0000039237  0000039360  0000039483  0000039606  0000039729  0000039852  0000039975  0000040098  0000040221  0000040344  0000040467  0000040590  0000040713  0000040836  0000040959  0000041082  0000041205  0000041328  0000041451  0000041574  0000041697  0000041820  0000041943  0000042066  0000042189  0000042312  0000042435  0000042558  0000042681  0000042804  0000042927  0000043050  0000043173  0000043296  0000043419  0000043542   0000043665  0000043788  0000043911  0000044034  0000044157  0000044280  0000044403  0000044526  0000044649  0000044772  0000044895  0000045018  0000045141  0000045264  0000045387   0000045510  0000045633  0000045756  0000045879  0000046002  0000046125  0000046248  0000046371  0000046494  0000046617  0000046740  0000046863  0000046986  0000047109  0000047232  0000047355  0000047478  0000047601  0000047724  0000047847  0000047970  ... This sequence number produces multiples of 123, beginning with 0 * 123, and writing them in 10 digit strings. Accurate to the 390th non-zero term (390 * 123 = 47970), which is the limit of my computation. This sequence is not in the OEIS.

We still have problem producing sequence numbers that end up showing multiples of some numbers.

You may have noticed that all of the sequence numbers that produce counting sequences end in a “1”.  Any number that ends in 1 cannot be divisible by 2 or 5.  So I did not show you a sequence number that shows multiples of 2 or 5 – yet.

I have developed two methods of dealing with this issue – but you may not consider them to be valid solutions.  For now just watch and learn.  Then you will have time to think about these methods and decide if you thing they are OK, or not OK.

Multiples of 2 2/998001 does not simplify.  However, 2/998001 can be written as the sum of two unit fractions 1/499001 and 1/498003497001.  (Which brings up the issue of using more than 1 fraction – is it fair?)  So let’s do it. 2/998001 = 1/499001 + 1/498003497001 = 0. 000  002  004  006  008  010  012  014  016  018  020  022  024  026  028  030  032  034  036  038  040  042  044  046  048  050  052  054  056  058  060  062  064  066  068  070  072  074  076  078  080  082  084  086  088  090  092  094  096  098  100  102  104  106  108  110  112  114  116  118  120  122  124  126  128  130  132  134  136  138  140  142  144  146  148  150  152  154  156  158  160  162  164  166  168  170  172  174  176  178  180  182  184  186  188  190  192  194  196  198  200  202  204  206  208  210  212  214  216  218  220  222  224  226  228  230  232  234  236  238  240  242  244  246  248  250  252  254  256  258  260  262  264  266  268  270  272  274  276  278  280  282  284  286  288  290  292  294  296  298  300  302  304  306  308  310  312  314  316  318  320   322 324  326  328  330  332  334  336  338  340  342  344  346  348  350  352  354  356  358  360  362  364  366  368  370  372  374  376  378  380  382  384  386  388  390  392  394  396  398  400  402  404  406  408  410  412  414  416  418  420  422  424  426  428  430  432  434  436  438  440  442  444  446  448  450  452  454  456  458  460  462  464  466  468  470  472  474  476  478  480  482  484  486  488  490  492  494  496  498  500  502  504  506  508  510  512  514  516  518  520  522  524  526  528  530  532  534  536  538  540  542  544  546  548  550  552  554  556  558  560  562  564  566  568  570  572  574  576  578  580  582  584  586  588  590  592  594  596  598  600  602  604  606  608  610  612  614  616  618  620  622  624  626  628  630  632  634  636  638  640  642  644  646  648  650  652  654  656  658  660  662  664  666  668  670  672  674  676  678  680  682  684  686  688  690  692  694  696  698  700  702  704  706  708  710  712  714  716  718  720  722  724  726  728  730  732  734  736  738  740  742  744  746  748  750  752  754  756  758  760  762  764  766  768  770  772  774  776  778  780  782  784  786  788  790  792  794  796  798  800  802  804  806  808  810  812  814  816  818  820  822  824  826  828  830  832  834  836  838  840  842  844  846  848  850  852  854  856  858  860  862  864  866  868  870  872  874  876  878  880  882  884  886  888  890  892  894  896  898  900  902  904  906  908  910  912  914  916  918  920  922  924  926  928  930  932  934  936  938  940  942  944  946  948  950  952  954  956  958  960  962  964  966  968  970  972  974  976  978  980  982  984  986  988  990  992  994  996  Written in three digit strings. Accurate to 498th non-zero term (996), then it skips 998, and it starts doing odd numbers.  (Note that 998 is the only one, two, or three digit even number that it skipped.) Compare with OEIS sequence A005843.

Multiples of 5: 5/998002 does not simplify, but it can be converted into three fractions that all have 1 as their numerator.  I can add the results of each of these fractions to get the results that I am looking for.  This means it is not at type 1 sequence number, but it is still mathematically kewl.  (“Kewl” is pronounced like “cool” and “kool”, but it does not refer to the temperature, a childs drink, or a brand of cigarettes.) 5/998001 = 1/199601 + 1/49800499401 + 1/3306786320735534645667 = 0. 000  005  010  015  020  025  030  035  040  045  050  055  060  065  070  075  080  085  090  095  100  105  110  115  120  125  130  135  140  145  150  155  160  165  170  175  180  185  190  195  200  205  210  215  220  225  230  235  240  245  250  255  260  265  270  275  280  285  290  295  300  305  310  315  320  325  330  335  340  345  350  355  360  365  370  375  380  385  390  395  400  405  410  415  420  425  430  435  440  445  450  455  460  465  470  475  480  485  490  495  500  505  510  515  520  525  530  535  540  545  550  555  560  565  570  575  580  585  590  595  600  605  610  615  620  625  630  635  640  645  650  655  660  665  670  675  680  685  690  695  700  705  710  715  720  725  730  735  740  745  750  755  760  765  770  775  780  785  790  795  800  805  810  815  820  825  830  835  840  845  850  855  860  865  870  875  880  885  890  895  900  905  910  915  920  925  930  935  940  945  950  955  960  965  970  975  980  985  990  … The sequence only skips one multiple of 5 (1, 2, or 3 digit multiples of 5). It also has an extra zero at the beginning of the decimal expansion.  It sticks out like a sore thumb. Written in three digit strings. Accurate up to 990, the 198th non-zero term. Compare with OEIS sequence A008587.

So at this point we know how to find a sequence number that will produce a decimal expansion that counts as high as we want one to count.  And we can find a sequence number that will show multiples of 3, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 53, 73, 79, 101, 137, 211, 239, 241, 271, 281, 757, 859 (and some larger primes), and some combinations of these numbers.  We also know that we can find ways to deal with some of the numbers that we can’t deal with using the method mentioned above. So right now you know more about how to find sequence numbers than … well, everybody that has not visited this site – which is almost everybody. Welcome to the Sequence Number Wizard Club! Stick around, check back later and you will learn more.

 


David