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- TotativeDigits was defined incorrectly, previously counted totatives
from 0 to r-1, now counts digits from 1 to r. This ensures that
len(TotativeDigits(r)) = Totient(r).
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This is intended to allow users to easily view the digit maps of many
different radices at once by running the program multiple times.
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Instead of just saying how many totative digits there are, the new -t
flag allows the user to determine the specific digits. All totatives
will end in one of these digits, and all primes are either factors or
totatives. This feature is not useful in comparing radices, only
learning one you have already chosen.
Because there can be so many totatives (p - 1 of them for primes!), this
is not displayed by default. The digit map (without -f) gives this
information, and beyond its range every number will have more than 8
totatives, so do not use this flag if you only want the count.
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factors.Type now supports all numbers; I have used lookup arrays instead
of determining whether a number is SAN or not. There are only 117
elements to store, and this makes the algorithm Θ(1), so it's an
improvement.
Also, I have changed the size of some integer values to correspond to
this change - they now indicate the size of numbers they can accept.
The only outputs that are hidden for large radices are:
- The digit map, which goes up to 36 because I don't have any more
digits beyond that point
- The multiplication table complexity, which is estimated above 2^16
(for performance), and can optionally be extended to 2^32 (above this,
the output could overflow a uint64).
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because it's not public API!
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These radices are large enough that:
- there is no reason to use them as actual radices
- calculating them takes a lot of time!
Therefore, the exact MTC and radix type shouldn't be calculated by
default. If you want to take the time, you still can with -l. I am
keeping the original 2^32 limit even with -l, because the problem with
that is not performance, it is that the resulting MTC could overflow a
uint64 (also the CAN list only goes up to this range).
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Full digit maps will show every digit from 0 to 35, regardless of the
radix. This allows you to see extra fractions for small radices and to
get a digit map for radices above 36.
This isn't enabled by default because the extra "digits" added for small
radices aren't actually digits - so it may be unintuitive. There are
also some situations where only a radix's actual digits matter, such as
multiplication tables.
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This achieves two things:
- Decouples my code by putting the printing code into its own file
- Makes it easier to make alternate ways of printing (e.g. a compact
mode)
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This is not in the output yet, but it will be soon - printing it is
another task since I want colours in my output.
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This ensures the output can fit into a uint64. Also, calculating it at
this stage is slow, and not calculating it can make the program nearly
instant even for very large numbers!
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This type measures which kind of classes each radix is a part of:
- Colossally Abundant (OEIS: A004490; factor score better than every
other number if you account for size be dividing by a certain power of
the number)
- Superabundant (OEIS: A004394; factor score better than every smaller
number)
- Ordered-Exponent (OEIS: A025487; exponents in prime factorization go
down as you get to bigger primes, and no prime is skipped)
- Practical (OEIS: A005153; factors can sum to any number below the
original number without duplication)
Each of these groups is a subset of the next, so only the most specific
label is reported.
The purpose of this program is to give you useful info to help you
determine which radices are the best, and these categories give a rough,
quantitative measure of how useful a radix's factors are:
- Practical is approximately the minimum requirement for a worthwhile
radix. Non-practical radices above ~16 are probably terrible to use.
- Ordered-Exponent radices act like local maxima - you can't get any
better (smaller) without changing the "shape" (exponents) of your prime
factorization.
- Superabundant radices are the best radices below the next
superabundant number (e.g. 12 is the best radix below 24).
- Colossally abundant radices are, in some sense, the best radices out of
all numbers.
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A radix's logarithm determines how well it compresses digits - a higher
logarithm means numbers will take up fewer digits. If c =
log(a)/log(b), then numbers in radix b will be around c times longer
than numbers in radix a.
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(MTC = Multiplication Table Complexity)
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Use methods from the new slices and maps modules to make my code faster
and more compact.
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