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 Base 168, The tetradecimal "short hundred"
Double sharp
Posted: Nov 5 2017, 07:59 AM


Dozens Disciple


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Joined: 19-September 15



Base 168

Base 168 is a “grand base” that lies well beyond our current ability to wield as a number base of general human arithmetic. Its multiplication table is astronomical (14,196 unique products, over 258 times the size of the decimal table) so any notion of memorizing it is beyond the ability of the average kid in school. Why consider such a gigantic number as a base?

Lovers of the great hundred may find 168 an extension of its excellent properties, simply swapping a factor of 5 for 7. The number 168 is another step in the sequence {1, 2, 3, 4, 6, 8, 10, 12, 18, 20, 24, 30, 36, 48, 60, 72, 84, 90, 96, 108, 120, 168, …}, the largely composite numbers (see OEIS A067128). These numbers set or equal previous records for their number of divisors. If we are interested in producing a highly patterned number base so we creatures sensitive to pattern can wield said number base, we want to maximize the divisors in a base.

Seen in the light of 120, 168 seems to be a reasonable tetradecimal analog. It keeps a beneficial indirect relationship with the square of a slightly larger prime, now 13 instead of 11, but sacrifices a relationship with 5 completely in favor of having a large prime as the omega. The number 168 does not have the sort of convenient neighbors on both sides like 120 does. Perhaps the very best feature is that 168 is less than the tetradecimal hundred, while 120 is more than the decimal hundred: thus the use of 168 does not run into the same sticking point of needing transdecimal figures that 120 does.

Base 168 is more difficult to wield than base 120. Its magnitude, being greater than that of 120, is already at the point where abbreviated-multiplication-table techniques seem totally infeasible; however, we could conceive of tetradecimally coding it as 12-on-14. Seen in this light, 84 and 168 could reasonably take their places as analogs to 60 and 120 in a tetradecimal world, although the increased difficulty of tetradecimal compared to decimal might hasten the decay of 168 to pure tetradecimal, to be used only as an auxiliary, where its high divisibility minimizes the need to turn to fractions but where its arithmetic is tamed by that of base 14.

Let’s take a look at 168 as a number base. Note that this examination is not as thorough as those for smaller bases, for obvious reasons.

(Please refer to “Icarus’s Standard Nomenclature for Number Bases” for the legend of the digit map below and any terminology. References to elementary number theory books are given in that post and thread to support what is written in this post.)

Base 168 has the following properties:

Digits of Base 168, using 12-on-14 or tetradecimal coded notation, primes in boldface type

Digit Map Divisibility Rules
  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d
0_ 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d
1_ 10 11 12 13 14 15 16 17 18 19 1a 1b 1c 1d
2_ 20 21 22 23 24 25 26 27 28 29 2a 2b 2c 2d
3_ 30 31 32 33 34 35 36 37 38 39 3a 3b 3c 3d
4_ 40 41 42 43 44 45 46 47 48 49 4a 4b 4c 4d
5_ 50 51 52 53 54 55 56 57 58 59 5a 5b 5c 5d
6_ 60 61 62 63 64 65 66 67 68 69 6a 6b 6c 6d
7_ 70 71 72 73 74 75 76 77 78 79 7a 7b 7c 7d
8_ 80 81 82 83 84 85 86 87 88 89 8a 8b 8c 8d
9_ 90 91 92 93 94 95 96 97 98 99 9a 9b 9c 9d
a_ a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 aa ab ac ad
b_ b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 ba bb bc bd
  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d
0_ 00 01 02 03 04 05 06 07 08 09 0a 0b 0c 0d
1_ 10 11 12 13 14 15 16 17 18 19 1a 1b 1c 1d
2_ 20 21 22 23 24 25 26 27 28 29 2a 2b 2c 2d
3_ 30 31 32 33 34 35 36 37 38 39 3a 3b 3c 3d
4_ 40 41 42 43 44 45 46 47 48 49 4a 4b 4c 4d
5_ 50 51 52 53 54 55 56 57 58 59 5a 5b 5c 5d
6_ 60 61 62 63 64 65 66 67 68 69 6a 6b 6c 6d
7_ 70 71 72 73 74 75 76 77 78 79 7a 7b 7c 7d
8_ 80 81 82 83 84 85 86 87 88 89 8a 8b 8c 8d
9_ 90 91 92 93 94 95 96 97 98 99 9a 9b 9c 9d
a_ a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 aa ab ac ad
b_ b0 b1 b2 b3 b4 b5 b6 b7 b8 b9 ba bb bc bd
  • Of the 39 primes p less than r = 168, only {2, 3, 7} are divisors of r. The remaining 36 primes q are coprime to r (i.e., the greatest common divisor (highest common factor) gcd(q, r) = 1).
  • There are 16 divisor digits d | r: {0, 1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84} (the numeral “0” signifying an arbitrary integer n ≡ 0 (mod 168).) Read this brief description of the divisor.
  • There are 48 totatives t such that gcd(t, r) = 1. Two-sevenths or 28.6% of the digits of base 168 are totatives (the totient ratio φ(r)/r = 2/7 for all bases r having the distinct prime divisors {2, 3, 7}). Read this brief definition and description of the significance of the totative.
  • Base 180 has 20 semidivisors (regular numbers gr that do not divide r evenly). This set includes any integer 0 < gr that is the product strictly of one or more of the distinct prime divisors of 168 that does not divide 168 evenly. Together with the 16 divisors, there are 36 regular digits in base 168; just over a fifth (21.4%) of the digits of base 168 are regular. This post defines regular numbers, and their significance.
  • There are 105 neutral digits in the scale of 168, including the 20 aforementioned semidivisors; the remaining 85 are semitotatives, products pq of at least one distinct prime divisor p and at least one prime q that is coprime to 168. Exactly 5/8 (62.5%) of the digits of base 168 are neutral; the preponderance are semitotatives, making up just over half (50.6%) of all the digits of base 168. The largely composite number 168 lies just over the line where semitotatives are the dominant type of digit. As the HCN gets larger, even more semicoprime numbers are admitted as digits, such that they dominate highly composite 7-smooth bases more intensively.
  • Let the integers ω = (r − 1) and α = (r + 1). The ω-totative of base 168 is 167, which is prime and does not distribute the omega divisibility rules to smaller digits. The “digit-sum” rule would be an academic curiosity in base 168 as it applies to digit-167, a large prime. The α-number of base 168 is 169 = 13². Digit-thirteen enjoys the “alpha” divisibility tests, i.e., the “alternating sum” rule, along with base 168’s “11” = 169. Multiples of either 13, 13² and any of the divisors of 168 enjoy intuitive compound divisibility tests. Thirteen is a fairly large prime, exceeding the decimal 11 for which the decimal alpha test applies. Observing that the decimal alternating-sum test is fairly obscure, one can say that any test for divisibility by 13 will be more so. See this post for maps and descriptions of the alpha divisibility rules for bases less than 30.
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Oschkar
Posted: Nov 5 2017, 08:02 AM


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Just a typo: OEIS A002201 is the SHCNs. The largely composites are A067128.
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Double sharp
Posted: Nov 5 2017, 08:08 AM


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QUOTE (Oschkar @ Nov 5 2017, 08:02 AM)
Just a typo: OEIS A002201 is the SHCNs. The largely composites are A067128.

Thanks! I copy-pasted some of the opening from 120 and 360, which explains the mistake. I've corrected it now.
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Double sharp
Posted: Nov 7 2017, 11:44 PM


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Group: Members
Posts: 1,402
Member No.: 1,150
Joined: 19-September 15



I am currently reconsidering the decision to show the digits in the maps in tetradecimal coding: it does accentuate the similarities to the long hundred but it is also not quite as easy to read. So here are similar maps with decimal digit values:
Digit Map Divisibility Rules
  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13
1_ 14 15 16 17 18 19 20 21 22 23 24 25 26 27
2_ 28 29 30 31 32 33 34 35 36 37 38 39 40 41
3_ 42 43 44 45 46 47 48 49 50 51 52 53 54 55
4_ 56 57 58 59 60 61 62 63 64 65 66 67 68 69
5_ 70 71 72 73 74 75 76 77 78 79 80 81 82 83
6_ 84 85 86 87 88 89 90 91 92 93 94 95 96 97
7_ 98 99 100 101 102 103 104 105 106 107 108 109 110 111
8_ 112 113 114 115 116 117 118 119 120 121 122 123 124 125
9_ 126 127 128 129 130 131 132 133 134 135 136 137 138 139
a_ 140 141 142 143 144 145 146 147 148 149 150 151 152 153
b_ 154 155 156 157 158 159 160 161 162 163 164 165 166 167
  _0 _1 _2 _3 _4 _5 _6 _7 _8 _9 _a _b _c _d
0_ 0 1 2 3 4 5 6 7 8 9 10 11 12 13
1_ 14 15 16 17 18 19 20 21 22 23 24 25 26 27
2_ 28 29 30 31 32 33 34 35 36 37 38 39 40 41
3_ 42 43 44 45 46 47 48 49 50 51 52 53 54 55
4_ 56 57 58 59 60 61 62 63 64 65 66 67 68 69
5_ 70 71 72 73 74 75 76 77 78 79 80 81 82 83
6_ 84 85 86 87 88 89 90 91 92 93 94 95 96 97
7_ 98 99 100 101 102 103 104 105 106 107 108 109 110 111
8_ 112 113 114 115 116 117 118 119 120 121 122 123 124 125
9_ 126 127 128 129 130 131 132 133 134 135 136 137 138 139
a_ 140 141 142 143 144 145 146 147 148 149 150 151 152 153
b_ 154 155 156 157 158 159 160 161 162 163 164 165 166 167
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