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Double sharp 
Posted: Nov 5 2017, 07:59 AM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 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 abbreviatedmultiplicationtable techniques seem totally infeasible; however, we could conceive of tetradecimally coding it as 12on14. 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 12on14 or tetradecimal coded notation, primes in boldface type


Oschkar 
Posted: Nov 5 2017, 08:02 AM

Dozens Disciple Group: Members Posts: 575 Member No.: 623 Joined: 19November 11 
Just a typo: OEIS A002201 is the SHCNs. The largely composites are A067128.

Double sharp 
Posted: Nov 5 2017, 08:08 AM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 15 
Thanks! I copypasted some of the opening from 120 and 360, which explains the mistake. I've corrected it now. 

Double sharp 
Posted: Nov 7 2017, 11:44 PM


Dozens Disciple Group: Members Posts: 1,402 Member No.: 1,150 Joined: 19September 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:

