Keith numbers in base 2

CraigD · 02-24-2008

Keith numbers are integers that appear in an n-Fibonacci sequence beginning with the digits of the base ten numeration of the number, where n is the number of digits in the numeral. For example, the 3-fibonachi sequence 7,4,2,13,19,34,66,119,219,404,742, ... has at its 11th term the number 742, matching its first 3 terms, 7,4,2.

An interesting thing about Keith numbers is that they are rare. For the range 10 to $10^{26}$ , it’s been exhaustively shown that there are only 84 of them. The largest known Keith number is currently 70267375510207885242218837404 (29 digits). There’s no efficient technique for generating Keith numbers – all known techniques involves exhaustively testing large collections of numbers.

Interesting unproven conjectures about Keith numbers are that there are infinitely many of them, or the negation, that there is a greatest Keith number.

Because Keith numbers depend on the choice of numeral base (usually 10), curiosity drove me to examine them in bases other than 10. For example, here are the Keith numbers under $1000_{10}$ in bases 2 to 16:

Code:

Base:         2        3      4     5     6     7     8     9   10   11   12   13   14   15   16
  2          10        .      .     .     .     .     .     .    .    .    .    .    .    .    .    2
  3          11       10      .     .     .     .     .     .    .    .    .    .    .    .    .    3
  4         100        .      .     .     .     .     .     .    .    .    .    .    .    .    .    4
  5           .       12     11    10     .     .     .     .    .    .    .    .    .    .    .    5
  6           .       20      .     .     .     .     .     .    .    .    .    .    .    .    .    6
  7           .       21     13     .     .     .     .     .    .    .    .    .    .    .    .    7
  8        1000        .      .     .    12    11    10     .    .    .    .    .    .    .    .    8
  9           .        .      .    14     .     .     .     .    .    .    .    .    .    .    .    9
 10           .        .     22    20     .     .     .     .    .    .    .    .    .    .    .   10
 11           .        .      .    21    15     .    13     .    .    .    .    .    .    .    .   11
 13           .        .      .    23     .    16     .     .    .   12   11   10    .    .    .   13
 14           .        .      .     .     .     .     .     .   14    .    .    .    .    .    .   14
 15           .        .     33    30     .     .    17     .    .    .    .    .    .    .    .   15
 16       10000        .      .     .    24    22    20     .    .    .    .    .    .    .    .   16
 17           .        .      .     .     .     .     .    18    .    .   15    .    .    .    .   17
 18           .        .    102     .     .     .     .     .    .    .    .    .    .   13    .   18
 19           .        .      .     .     .    25     .     .   19    .    .    .    .    .    .   19
 20           .        .      .    40     .     .     .     .    .    .    .    .   16    .    .   20
 21           .        .      .     .     .     .     .    23    .   1a    .    .    .    .    .   21
 22           .        .      .    42     .     .    26     .    .    .    .    .    .    .    .   22
 23           .        .      .     .     .     .     .     .    .    .   1b    .    .    .   17   23
 24           .        .      .     .     .    33    30     .    .    .    .    .    .    .    .   24
 25           .        .      .     .     .     .     .    27    .    .    .   1c    .    .    .   25
 26           .        .      .     .     .     .     .     .    .   24   22   20    .    .    .   26
 27           .        .      .     .    43     .     .     .    .    .    .    .   1d    .    .   27
 28           .        .      .     .     .     .     .     .   28    .    .    .    .    .    .   28
 29           .        .    131     .     .     .     .     .    .    .    .    .   21   1e    .   29
 31           .        .      .   111     .     .     .     .    .   29    .   25    .    .   1f   31
 32      100000        .      .     .     .    44    40     .    .    .    .    .    .    .    .   32
 34           .        .      .     .     .     .     .     .    .    .   2a    .    .    .    .   34
 36           .        .      .     .     .     .     .     .    .    .    .    .    .   26    .   36
 37           .        .      .     .   101     .    45     .    .    .   31   2b    .    .    .   37
 39           .        .      .     .     .     .     .     .    .   36   33   30    .    .    .   39
 40           .        .      .   130     .    55    50     .    .    .    .    .   2c    .    .   40
 42           .        .      .     .     .     .     .    46    .    .    .    .    .    .    .   42
 43           .        .      .   133     .     .     .     .    .    .    .    .    .   2d    .   43
 44           .        .      .     .   112     .     .     .    .    .    .    .    .    .    .   44
 45           .        .      .     .     .     .     .     .    .   41    .    .    .    .    .   45
 46           .        .      .     .     .     .     .     .    .    .    .    .    .    .   2e   46
 47           .        .    233     .     .     .     .     .   47    .    .    .    .    .    .   47
 48           .        .      .     .     .    66    60     .    .    .    .    .    .    .    .   48
 50           .        .      .     .     .     .     .     .    .    .    .    .    .    .   32   50
 52           .        .      .     .     .     .     .     .    .   48   44   40    .    .    .   52
 53           .        .      .   203     .     .     .     .    .    .    .    .    .    .    .   53
 54           .        .      .     .     .     .     .     .    .    .    .    .    .   39    .   54
 56           .        .      .     .     .     .    70     .    .    .    .    .    .    .    .   56
 57           .     2010      .     .     .   111     .     .    .    .   49    .    .    .    .   57
 58           .        .      .     .     .     .     .     .    .    .    .    .   42    .    .   58
 59           .        .      .     .     .     .    73     .    .    .    .    .    .    .    .   59
 61           .        .      .     .     .     .     .     .   61    .    .    .    .    .    .   61
 62           .        .      .   222     .     .     .     .    .    .    .   4a    .    .    .   62
 64     1000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .   64
 65           .        .      .     .     .     .     .     .    .   5a   55   50    .    .    .   65
 67           .        .      .     .     .     .     .    74    .    .    .    .   4b    .    .   67
 71           .        .      .   241     .     .     .     .    .    .    .    .    .    .    .   71
 72           .        .      .     .     .     .     .     .    .    .    .    .    .   4c    .   72
 74           .        .      .     .   202     .     .     .    .    .   62    .    .    .    .   74
 75           .        .      .     .     .     .     .     .   75    .    .    .    .    .    .   75
 77           .        .      .     .     .     .     .     .    .    .    .    .    .    .   4d   77
 78           .        .      .     .     .     .     .     .    .    .   66   60    .    .    .   78
 81           .        .      .     .     .     .     .   100    .    .    .    .    .    .    .   81
 83           .        .      .     .     .     .     .     .    .   76    .    .    .    .    .   83
 84           .        .      .   314     .     .     .     .    .    .    .    .    .    .    .   84
 87           .        .      .     .     .     .     .     .    .    .    .    .   63    .    .   87
 88           .        .      .     .   224     .     .     .    .    .    .    .    .    .    .   88
 90           .        .      .     .     .     .     .     .    .   82    .    .    .    .    .   90
 91           .        .      .     .     .     .     .     .    .    .   77   70    .    .    .   91
 92           .        .      .     .     .     .   134     .    .    .    .    .    .    .    .   92
 93           .        .      .   333     .     .     .     .    .    .    .    .    .    .    .   93
 96           .        .      .     .     .     .     .   116    .    .    .    .    .    .    .   96
 99           .        .      .     .     .     .     .     .    .    .    .   78    .    .    .   99
100           .        .      .     .     .     .     .     .    .    .    .    .    .    .   64  100
101           .        .      .     .     .     .     .   122    .    .    .    .    .    .    .  101
102           .    10210      .     .     .     .     .     .    .    .    .    .    .    .    .  102
104           .        .      .     .     .     .     .     .    .    .   88   80    .    .    .  104
107           .        .      .     .     .     .     .     .    .    .    .    .   79    .    .  107
111           .        .      .     .   303     .     .     .    .    .   93    .    .    .    .  111
113           .        .   1301     .     .     .     .     .    .    .    .    .    .    .    .  113
114           .        .      .     .     .   222     .     .    .    .    .    .    .    .    .  114
115           .        .      .     .     .     .     .     .    .    .    .    .    .   7a    .  115
116           .        .      .     .     .     .     .     .    .    .    .    .   84    .    .  116
117           .        .      .     .     .     .     .     .    .    .   99   90    .    .    .  117
123           .        .      .     .     .     .   173     .    .    .    .    .    .    .   7b  123
124           .        .      .   444     .     .     .     .    .    .    .    .    .    .    .  124
125           .        .      .     .     .   236     .     .    .    .    .    .    .    .    .  125
127           .    11201      .     .     .     .     .     .    .    .    .    .    .    .    .  127
128    10000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .  128
130           .        .      .     .     .     .     .     .    .    .   aa   a0    .    .    .  130
143    10001111        .      .     .     .     .     .     .    .    .   bb   b0    .    .    .  143
145           .        .      .     .     .   265     .     .    .    .    .    .   a5    .    .  145
148           .        .      .     .   404     .     .     .    .    .    .    .    .    .    .  148
149           .        .      .     .     .     .     .   175    .    .    .    .    .    .    .  149
150           .        .      .     .     .     .     .     .    .    .    .    .    .    .   96  150
151           .        .      .     .     .     .     .     .    .    .    .    .    .   a1    .  151
154           .        .      .  1104     .     .     .     .    .    .    .    .    .    .    .  154
156           .        .      .     .     .     .     .     .    .    .    .   c0    .    .    .  156
161           .        .      .     .     .     .     .     .    .    .    .   c5    .    .    .  161
162           .        .      .     .     .     .     .   200    .    .    .    .    .    .    .  162
163           .        .   2203     .     .     .     .     .    .    .    .    .    .    .    .  163
171           .        .      .     .     .   333     .     .    .    .    .    .    .    .    .  171
173           .        .      .     .     .     .     .   212    .    .  125    .    .    .    .  173
174           .        .      .     .     .     .     .     .    .    .    .    .   c6    .    .  174
185           .        .      .     .   505     .     .     .    .    .    .    .    .    .    .  185
187           .        .      .     .     .     .     .     .    .    .    .    .    .   c7    .  187
197           .        .      .     .     .     .     .     .  197    .    .    .    .    .    .  197
200           .        .      .     .     .     .   310     .    .    .    .    .  104    .   c8  200
202           .        .      .     .     .     .     .   244    .    .    .    .    .    .    .  202
206           .    21122      .     .     .     .     .     .    .    .    .    .    .    .    .  206
217           .    22001      .     .     .     .     .     .    .    .    .    .    .    .    .  217
221           .        .      .  1341     .     .     .     .    .    .    .    .    .    .    .  221
228           .        .      .     .     .   444     .     .    .    .    .    .    .    .    .  228
243           .        .      .     .     .     .     .   300    .    .    .    .    .    .    .  243
249           .        .      .     .     .     .     .     .    .    .    .    .    .  119    .  249
250           .        .      .     .     .     .     .     .    .    .    .    .    .    .   fa  250
251           .        .      .     .     .     .   373     .    .    .    .    .    .    .    .  251
256   100000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .  256
257           .        .      .     .     .     .   401     .    .    .    .    .    .    .    .  257
262           .        .      .     .     .     .     .     .    .  219    .    .    .    .    .  262
269           .        .  10031     .     .     .     .     .    .    .    .    .    .    .    .  269
274           .        .      .     .     .     .     .     .    .    .    .    .    .    .  112  274
285   100011101        .      .     .     .   555     .     .    .    .    .    .    .    .    .  285
303           .        .      .     .     .     .     .   366    .    .    .    .    .    .    .  303
305           .        .      .     .     .     .     .     .    .    .  215    .    .    .    .  305
324           .        .      .     .     .     .     .   400    .    .    .    .    .    .    .  324
329           .        .      .     .     .   650     .     .    .    .    .    .    .    .    .  329
342           .        .      .     .     .   666     .     .    .    .    .    .    .    .    .  342
346           .        .      .     .     .     .     .   424    .    .  24a    .    .    .    .  346
400           .        .      .     .     .     .   620     .    .    .    .    .  208    .    .  400
401           .        .      .     .     .     .     .     .    .    .    .    .    .    .  191  401
404           .        .      .     .     .     .     .   488    .    .    .    .    .    .    .  404
405           .        .      .     .     .     .     .   500    .    .    .    .    .    .    .  405
409           .        .      .     .  1521     .     .     .    .    .    .    .    .    .  199  409
457           .        .      .     .     .     .   711     .    .    .    .    .    .    .    .  457
483           .        .      .  3413     .     .     .     .    .    .    .    .    .    .    .  483
486           .        .      .     .     .     .     .   600    .    .    .    .    .    .    .  486
512  1000000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .  512
519           .        .      .     .     .     .     .   636    .    .    .    .    .    .    .  519
524           .        .      .     .     .     .     .     .    .    .    .  314    .    .    .  524
526           .        .      .     .  2234     .     .     .    .    .    .    .    .    .    .  526
528           .        .      .     .     .     .     .     .    .    .    .    .    .    .  210  528
529           .        .      .     .     .     .     .     .    .  441    .    .    .    .    .  529
542           .        .      .     .     .     .     .     .    .    .    .  329    .    .    .  542
545           .        .      .     .     .     .     .     .    .  456    .    .    .    .    .  545
548           .        .      .     .     .     .     .     .    .    .    .    .    .    .  224  548
567           .        .      .     .     .     .     .   700    .    .    .    .    .    .    .  567
569  1000111001        .      .     .     .     .     .     .    .    .    .    .    .    .    .  569
581           .        .      .     .     .     .     .     .    .    .  405    .    .    .    .  581
589           .        .      .     .     .  1501     .     .    .    .    .    .    .    .    .  589
600           .        .      .     .     .     .     .     .    .    .    .    .  30c    .    .  600
610           .        .      .     .     .     .     .     .    .    .  42a    .    .    .    .  610
647           .        .      .     .     .     .     .     .    .    .    .    .  343    .    .  647
648           .        .      .     .     .     .     .   800    .    .    .    .    .    .    .  648
677           .   221002      .     .     .     .     .     .    .    .    .    .    .    .    .  677
683  1010101011        .      .     .     .     .     .     .    .    .    .    .    .    .    .  683
692           .        .      .     .     .     .     .   848    .    .    .    .    .    .    .  692
718           .        .      .     .     .     .     .     .    .    .    .    .    .  32d    .  718
732           .        .      .     .     .     .     .  1003    .    .    .    .    .    .    .  732
742           .        .      .     .     .     .     .     .  742    .    .    .    .    .    .  742
805           .  1002211      .     .     .     .     .     .    .    .    .    .    .    .    .  805
822           .        .      .     .     .     .     .     .    .    .    .    .    .    .  336  822
840           .  1011010      .     .     .     .     .     .    .    .    .    .    .    .    .  840
857           .        .      .     .     .     .     .  1152    .    .    .    .    .    .    .  857
893           .        .      .     .     .     .  1575     .    .    .    .    .    .    .    .  893
928           .        .      .     .     .     .     .     .    .    .  654    .    .    .    .  928
939           .        .      .     .     .     .     .     .    .    .    .  573    .    .    .  939
Base:         2        3      4     5     6     7     8     9   10   11   12   13   14   15   16
Count(<1000):14       12     10    20    13    18    20    25    8   13   24   21   15   11   16

Although there are fewer Keith numbers < 1000 in base 10 than the other first 15 bases (8 vs. the max of 25 for base 9), this doesn’t appear to be a trend when more numbers are considered. When Keith numbers < 100000 are counted for bases 2 through 32, 100, and 1000, 24 are found for base 10, vs. a minimum of 16 in base 4 and a maximum of 46 in base 21.

One of the first remarkable observations to come from my foray into Keith numbers in different bases is that, for base 2, every power of 2 is a Keith number (but every Keith number is not a power of 2). A proof of this is trivial (and left as an exercise for the reader

), so the conjecture that there are infinitely many Keith numbers is proven true in base 2.

Nothing similarly obvious jumps out at me for other bases. However, I wonder if this obvious, trivial proof holds an intuitive gateway to a significant one involving the infinite number conjecture in bases greater than 2?

Anybody been down this road? If not, consider it new food for thought for discrete math fans.

Turtle · 02-26-2008

Quote:

Originally Posted by CraigD

Keith numbers are integers that appear in an n-Fibonacci sequence beginning with the digits of the base ten numeration of the number, where n is the number of digits in the numeral. For example, the 3-fibonachi sequence 7,4,2,13,19,34,66,119,219,404,742, ... has at its 11th term the number 731, matching its first 3 terms, 7,3,1.

Mmmmm...It looks to me like the 11th term is 742 in the example, and that the first 3 terms are 7, 4, 2? What am I missing?

Quote:

Originally Posted by Craig

Anybody been down this road? If not, consider it new food for thought for discrete math fans.

Is it a yellow brick road? Mmmmm....tasty musings. I'm going to have to chew the gristly list a bit as I'm curious what intersecton with other familiar sets so succinctly served Keith numbers may enjoy. ...

Turtle · 02-27-2008

Quote:

Originally Posted by CraigD

Because Keith numbers depend on the choice of numeral base (usually 10), curiosity drove me to examine them in bases other than 10. For example, here are the Keith numbers under $1000_{10}$ in bases 2 to 16:

Code:

Base:         2        3      4     5     6     7     8     9   10   11   12   13   14   15   16
  2          10        .      .     .     .     .     .     .    .    .    .    .    .    .    .    2
  3          11       10      .     .     .     .     .     .    .    .    .    .    .    .    .    3
  4         100        .      .     .     .     .     .     .    .    .    .    .    .    .    .    4
  5           .       12     11    10     .     .     .     .    .    .    .    .    .    .    .    5
  6           .       20      .     .     .     .     .     .    .    .    .    .    .    .    .    6
  7           .       21     13     .     .     .     .     .    .    .    .    .    .    .    .    7
  8        1000        .      .     .    12    11    10     .    .    .    .    .    .    .    .    8
  9           .        .      .    14     .     .     .     .    .    .    .    .    .    .    .    9
 10           .        .     22    20     .     .     .     .    .    .    .    .    .    .    .   10
 11           .        .      .    21    15     .    13     .    .    .    .    .    .    .    .   11
 13           .        .      .    23     .    16     .     .    .   12   11   10    .    .    .   13
 14           .        .      .     .     .     .     .     .   14    .    .    .    .    .    .   14
 15           .        .     33    30     .     .    17     .    .    .    .    .    .    .    .   15
 16       10000        .      .     .    24    22    20     .    .    .    .    .    .    .    .   16
 17           .        .      .     .     .     .     .    18    .    .   15    .    .    .    .   17
 18           .        .    102     .     .     .     .     .    .    .    .    .    .   13    .   18
 19           .        .      .     .     .    25     .     .   19    .    .    .    .    .    .   19
 20           .        .      .    40     .     .     .     .    .    .    .    .   16    .    .   20
 21           .        .      .     .     .     .     .    23    .   1a    .    .    .    .    .   21
 22           .        .      .    42     .     .    26     .    .    .    .    .    .    .    .   22
 23           .        .      .     .     .     .     .     .    .    .   1b    .    .    .   17   23
 24           .        .      .     .     .    33    30     .    .    .    .    .    .    .    .   24
 25           .        .      .     .     .     .     .    27    .    .    .   1c    .    .    .   25
 26           .        .      .     .     .     .     .     .    .   24   22   20    .    .    .   26
 27           .        .      .     .    43     .     .     .    .    .    .    .   1d    .    .   27
 28           .        .      .     .     .     .     .     .   28    .    .    .    .    .    .   28
 29           .        .    131     .     .     .     .     .    .    .    .    .   21   1e    .   29
 31           .        .      .   111     .     .     .     .    .   29    .   25    .    .   1f   31
 32      100000        .      .     .     .    44    40     .    .    .    .    .    .    .    .   32
 34           .        .      .     .     .     .     .     .    .    .   2a    .    .    .    .   34
 36           .        .      .     .     .     .     .     .    .    .    .    .    .   26    .   36
 37           .        .      .     .   101     .    45     .    .    .   31   2b    .    .    .   37
 39           .        .      .     .     .     .     .     .    .   36   33   30    .    .    .   39
 40           .        .      .   130     .    55    50     .    .    .    .    .   2c    .    .   40
 42           .        .      .     .     .     .     .    46    .    .    .    .    .    .    .   42
 43           .        .      .   133     .     .     .     .    .    .    .    .    .   2d    .   43
 44           .        .      .     .   112     .     .     .    .    .    .    .    .    .    .   44
 45           .        .      .     .     .     .     .     .    .   41    .    .    .    .    .   45
 46           .        .      .     .     .     .     .     .    .    .    .    .    .    .   2e   46
 47           .        .    233     .     .     .     .     .   47    .    .    .    .    .    .   47
 48           .        .      .     .     .    66    60     .    .    .    .    .    .    .    .   48
 50           .        .      .     .     .     .     .     .    .    .    .    .    .    .   32   50
 52           .        .      .     .     .     .     .     .    .   48   44   40    .    .    .   52
 53           .        .      .   203     .     .     .     .    .    .    .    .    .    .    .   53
 54           .        .      .     .     .     .     .     .    .    .    .    .    .   39    .   54
 56           .        .      .     .     .     .    70     .    .    .    .    .    .    .    .   56
 57           .     2010      .     .     .   111     .     .    .    .   49    .    .    .    .   57
 58           .        .      .     .     .     .     .     .    .    .    .    .   42    .    .   58
 59           .        .      .     .     .     .    73     .    .    .    .    .    .    .    .   59
 61           .        .      .     .     .     .     .     .   61    .    .    .    .    .    .   61
 62           .        .      .   222     .     .     .     .    .    .    .   4a    .    .    .   62
 64     1000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .   64
 65           .        .      .     .     .     .     .     .    .   5a   55   50    .    .    .   65
 67           .        .      .     .     .     .     .    74    .    .    .    .   4b    .    .   67
 71           .        .      .   241     .     .     .     .    .    .    .    .    .    .    .   71
 72           .        .      .     .     .     .     .     .    .    .    .    .    .   4c    .   72
 74           .        .      .     .   202     .     .     .    .    .   62    .    .    .    .   74
 75           .        .      .     .     .     .     .     .   75    .    .    .    .    .    .   75
 77           .        .      .     .     .     .     .     .    .    .    .    .    .    .   4d   77
 78           .        .      .     .     .     .     .     .    .    .   66   60    .    .    .   78
 81           .        .      .     .     .     .     .   100    .    .    .    .    .    .    .   81
 83           .        .      .     .     .     .     .     .    .   76    .    .    .    .    .   83
 84           .        .      .   314     .     .     .     .    .    .    .    .    .    .    .   84
 87           .        .      .     .     .     .     .     .    .    .    .    .   63    .    .   87
 88           .        .      .     .   224     .     .     .    .    .    .    .    .    .    .   88
 90           .        .      .     .     .     .     .     .    .   82    .    .    .    .    .   90
 91           .        .      .     .     .     .     .     .    .    .   77   70    .    .    .   91
 92           .        .      .     .     .     .   134     .    .    .    .    .    .    .    .   92
 93           .        .      .   333     .     .     .     .    .    .    .    .    .    .    .   93
 96           .        .      .     .     .     .     .   116    .    .    .    .    .    .    .   96
 99           .        .      .     .     .     .     .     .    .    .    .   78    .    .    .   99
100           .        .      .     .     .     .     .     .    .    .    .    .    .    .   64  100
101           .        .      .     .     .     .     .   122    .    .    .    .    .    .    .  101
102           .    10210      .     .     .     .     .     .    .    .    .    .    .    .    .  102
104           .        .      .     .     .     .     .     .    .    .   88   80    .    .    .  104
107           .        .      .     .     .     .     .     .    .    .    .    .   79    .    .  107
111           .        .      .     .   303     .     .     .    .    .   93    .    .    .    .  111
113           .        .   1301     .     .     .     .     .    .    .    .    .    .    .    .  113
114           .        .      .     .     .   222     .     .    .    .    .    .    .    .    .  114
115           .        .      .     .     .     .     .     .    .    .    .    .    .   7a    .  115
116           .        .      .     .     .     .     .     .    .    .    .    .   84    .    .  116
117           .        .      .     .     .     .     .     .    .    .   99   90    .    .    .  117
123           .        .      .     .     .     .   173     .    .    .    .    .    .    .   7b  123
124           .        .      .   444     .     .     .     .    .    .    .    .    .    .    .  124
125           .        .      .     .     .   236     .     .    .    .    .    .    .    .    .  125
127           .    11201      .     .     .     .     .     .    .    .    .    .    .    .    .  127
128    10000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .  128
130           .        .      .     .     .     .     .     .    .    .   aa   a0    .    .    .  130
143    10001111        .      .     .     .     .     .     .    .    .   bb   b0    .    .    .  143
145           .        .      .     .     .   265     .     .    .    .    .    .   a5    .    .  145
148           .        .      .     .   404     .     .     .    .    .    .    .    .    .    .  148
149           .        .      .     .     .     .     .   175    .    .    .    .    .    .    .  149
150           .        .      .     .     .     .     .     .    .    .    .    .    .    .   96  150
151           .        .      .     .     .     .     .     .    .    .    .    .    .   a1    .  151
154           .        .      .  1104     .     .     .     .    .    .    .    .    .    .    .  154
156           .        .      .     .     .     .     .     .    .    .    .   c0    .    .    .  156
161           .        .      .     .     .     .     .     .    .    .    .   c5    .    .    .  161
162           .        .      .     .     .     .     .   200    .    .    .    .    .    .    .  162
163           .        .   2203     .     .     .     .     .    .    .    .    .    .    .    .  163
171           .        .      .     .     .   333     .     .    .    .    .    .    .    .    .  171
173           .        .      .     .     .     .     .   212    .    .  125    .    .    .    .  173
174           .        .      .     .     .     .     .     .    .    .    .    .   c6    .    .  174
185           .        .      .     .   505     .     .     .    .    .    .    .    .    .    .  185
187           .        .      .     .     .     .     .     .    .    .    .    .    .   c7    .  187
197           .        .      .     .     .     .     .     .  197    .    .    .    .    .    .  197
200           .        .      .     .     .     .   310     .    .    .    .    .  104    .   c8  200
202           .        .      .     .     .     .     .   244    .    .    .    .    .    .    .  202
206           .    21122      .     .     .     .     .     .    .    .    .    .    .    .    .  206
217           .    22001      .     .     .     .     .     .    .    .    .    .    .    .    .  217
221           .        .      .  1341     .     .     .     .    .    .    .    .    .    .    .  221
228           .        .      .     .     .   444     .     .    .    .    .    .    .    .    .  228
243           .        .      .     .     .     .     .   300    .    .    .    .    .    .    .  243
249           .        .      .     .     .     .     .     .    .    .    .    .    .  119    .  249
250           .        .      .     .     .     .     .     .    .    .    .    .    .    .   fa  250
251           .        .      .     .     .     .   373     .    .    .    .    .    .    .    .  251
256   100000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .  256
257           .        .      .     .     .     .   401     .    .    .    .    .    .    .    .  257
262           .        .      .     .     .     .     .     .    .  219    .    .    .    .    .  262
269           .        .  10031     .     .     .     .     .    .    .    .    .    .    .    .  269
274           .        .      .     .     .     .     .     .    .    .    .    .    .    .  112  274
285   100011101        .      .     .     .   555     .     .    .    .    .    .    .    .    .  285
303           .        .      .     .     .     .     .   366    .    .    .    .    .    .    .  303
305           .        .      .     .     .     .     .     .    .    .  215    .    .    .    .  305
324           .        .      .     .     .     .     .   400    .    .    .    .    .    .    .  324
329           .        .      .     .     .   650     .     .    .    .    .    .    .    .    .  329
342           .        .      .     .     .   666     .     .    .    .    .    .    .    .    .  342
346           .        .      .     .     .     .     .   424    .    .  24a    .    .    .    .  346
400           .        .      .     .     .     .   620     .    .    .    .    .  208    .    .  400
401           .        .      .     .     .     .     .     .    .    .    .    .    .    .  191  401
404           .        .      .     .     .     .     .   488    .    .    .    .    .    .    .  404
405           .        .      .     .     .     .     .   500    .    .    .    .    .    .    .  405
409           .        .      .     .  1521     .     .     .    .    .    .    .    .    .  199  409
457           .        .      .     .     .     .   711     .    .    .    .    .    .    .    .  457
483           .        .      .  3413     .     .     .     .    .    .    .    .    .    .    .  483
486           .        .      .     .     .     .     .   600    .    .    .    .    .    .    .  486
512  1000000000        .      .     .     .     .     .     .    .    .    .    .    .    .    .  512
519           .        .      .     .     .     .     .   636    .    .    .    .    .    .    .  519
524           .        .      .     .     .     .     .     .    .    .    .  314    .    .    .  524
526           .        .      .     .  2234     .     .     .    .    .    .    .    .    .    .  526
528           .        .      .     .     .     .     .     .    .    .    .    .    .    .  210  528
529           .        .      .     .     .     .     .     .    .  441    .    .    .    .    .  529
542           .        .      .     .     .     .     .     .    .    .    .  329    .    .    .  542
545           .        .      .     .     .     .     .     .    .  456    .    .    .    .    .  545
548           .        .      .     .     .     .     .     .    .    .    .    .    .    .  224  548
567           .        .      .     .     .     .     .   700    .    .    .    .    .    .    .  567
569  1000111001        .      .     .     .     .     .     .    .    .    .    .    .    .    .  569
581           .        .      .     .     .     .     .     .    .    .  405    .    .    .    .  581
589           .        .      .     .     .  1501     .     .    .    .    .    .    .    .    .  589
600           .        .      .     .     .     .     .     .    .    .    .    .  30c    .    .  600
610           .        .      .     .     .     .     .     .    .    .  42a    .    .    .    .  610
647           .        .      .     .     .     .     .     .    .    .    .    .  343    .    .  647
648           .        .      .     .     .     .     .   800    .    .    .    .    .    .    .  648
677           .   221002      .     .     .     .     .     .    .    .    .    .    .    .    .  677
683  1010101011        .      .     .     .     .     .     .    .    .    .    .    .    .    .  683
692           .        .      .     .     .     .     .   848    .    .    .    .    .    .    .  692
718           .        .      .     .     .     .     .     .    .    .    .    .    .  32d    .  718
732           .        .      .     .     .     .     .  1003    .    .    .    .    .    .    .  732
742           .        .      .     .     .     .     .     .  742    .    .    .    .    .    .  742
805           .  1002211      .     .     .     .     .     .    .    .    .    .    .    .    .  805
822           .        .      .     .     .     .     .     .    .    .    .    .    .    .  336  822
840           .  1011010      .     .     .     .     .     .    .    .    .    .    .    .    .  840
857           .        .      .     .     .     .     .  1152    .    .    .    .    .    .    .  857
893           .        .      .     .     .     .  1575     .    .    .    .    .    .    .    .  893
928           .        .      .     .     .     .     .     .    .    .  654    .    .    .    .  928
939           .        .      .     .     .     .     .     .    .    .    .  573    .    .    .  939
Base:         2        3      4     5     6     7     8     9   10   11   12   13   14   15   16
Count(<1000):14       12     10    20    13    18    20    25    8   13   24   21   15   11   16

Some of the bases, as far as they go, have no pairs; that is to say elements that differ by 1. This is true of bases 4, 6, 10, 11, 14, 15, and 16. The opposite is true for complements, for example in base 13 we have the pair {1c,20}.

It is interesting to note that in base 10, the 3rd Keith number is the 2nd Perfect number,28, and the 1st Keith number is half of that, 14. Values seem of little value here when we cross bases, as for example no Keith number in base 8 is the 2nd Perfect number.

CraigD · 02-27-2008

Quote:

Originally Posted by Turtle

Mmmmm...It looks to me like the 11th term is 742 in the example, and that the first 3 terms are 7, 4, 2? What am I missing?

You’ve missed nothing but another example of my typing and proofreading incompetence

I’ve fixed it now.

Where I’m hoping to go in this thread is toward a rule for constructing Keith numbers in bases greater than 2. The rule in base 2 is trivially simple: “1 followed by 1 or more 0s”. Could there be a less simple rule for the other bases? Such a rule is also a proof of the “infinitely many” conjecture.

Turtle · 02-28-2008

Quote:

Originally Posted by CraigD

You’ve missed nothing but another example of my typing and proofreading incompetence

I’ve fixed it now.

Where I’m hoping to go in this thread is toward a rule for constructing Keith numbers in bases greater than 2. The rule in base 2 is trivially simple: “1 followed by 1 or more 0s”. Could there be a less simple rule for the other bases? Such a rule is also a proof of the “infinitely many” conjecture.

Roger the typo. To follow up, 731 is not Keith: {7 3 1 11 15 27 53 95 175 323 593 1091...}

I suspect no such rule exists, but it's an interesting conjecture.

The description you give on base 2 rule put me in mind of the similar base 2 katabataks triviality, wherein all values input to the function return 1, or if expressed as residues, 0.

Looking at the list with katabatak glasses, I see no pattern in any of the bases to suggest a rule either.

Keith numbers in base 2

Hypography?

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