Which is the expresssion of all "factorions" in the "factoradic" numerical base?

Question

Jay, with any search engine you'll find at the top of the list what a factorion is.

Answer 1

In the "factoradic" numbering system, the rightmost digit can be 0 or 1; the next digit to the left can be 0, 1, or 2; the next can be 0, 1, 2, or 3; and so on. That means that the rightmost digit counts ones, the next to the left counts sixes, the next one 24s, and so on.

[EDITED FROM THIS POINT FORWARD] A factorion is an integer which is equal to the sum of factorials of its digits. There are exactly four such numbers:

1! = 1 (decimal) = 1 (factoradic)
2! = 2 (decimal) = 10 (factoradic)
1! + 4! + 5! = 145 (decimal) = 11001 (factoradic) (note that the 1st, 4th, and 5th digits are all 1, the other digits are zero)
4! + 0! + 5! + 8! + 5! = 40585 (decimal) = 10021001 (you can work this one out :) )

For more information, see the linked articles below. Good luck!

[EDITED TO CORRECT] tbolling is right, I mistyped the above. The values of the factoradic digits from right to left are 1!=1, 2!=2, 3!=6, 4!=24, etc.

Answer 2

I agree with everything Jay H has to say with one small exception. In the first paragraph toward the end, Jay says that the 2nd place digit counts the sixes. The 2nd place digit in this numbering system counts the twos.

1! = 1
2! = 2
3! = 6
4! = 24
...

So, the first place digit counts the ones, the second place digit counts the twos, the third counts the sixes, etc.

Answer 3

No number can be represented by more than one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:

\sum_{i=0}^{n} i\cdot i! = {(n+1)!} - 1.

There is a natural mapping between the integers 0, ..., n! − 1 and permutations of n elements in lexicographic order, when the integers are expressed in factoradic form.

Answer 4

Mixed radix numeral system A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner....

[For more, click on this link]s are more general than the usual ones in that the numerical base may vary from position to position. Such numerical representation is advantageous when representing units that are equivalent to each other, but not by the same ratio. For example, 32 weeks, 5 days, 7 hours, 45 minutes, 15 seconds, and 500 milliseconds might be rendered relative to minutes in mixed-radix notation as:

... 3, 2, 5, 7, 45; 15, 500 , or as
... 10, 10, 7, 24, 60; 60, 1000

310210577244560.15605001000

In the tabular format, the digits are written above their base, and a semicolon is used to indicate the radix point[Click link for more facts about this topic]. In numeral format, each digit has its associated base attached as a subscript, and the radix point[For more, click on this link]'s position is indicated by a full stop A punctuation mark (.) placed at the end of a declarative sentence to indicate a full stop or after abbreviations

[For more, click on this link].

An MRN system can often benefit from a tabular summary. The familiar system for describing the 604800 seconds of a week starting from Sunday Midnight runs as follows:

Radix: 7 2 12 60 60
Denomination: day half-day hour minute second
Place value (seconds): 86400 43200 3600 60 1

Digit translations …
day: 0–Sunday 1–Monday 2–Tuesday 3–Wednesday 4–Thursday 5–Friday 6–Saturday
half-day: 0–AM 1–PM
hour: 0 is written as "12" (!)

So the MRN 371251251605760 seconds (from Midnight Sunday) is interpreted as 05:51:57 PM Wednesday, 070201202602460 as 12:02:24 AM Sunday. Ad-hoc notations for MRN systems are commonplace.

A second example of a mixed radix numeral system A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner....

[For more facts about this topic, click this link] in current use is in the design and use of currency The metal or paper medium of exchange that is presently used

[For more info, click on this link], where a limited set of denominations are minted with the objective of being able to represent any monetary quantity; the amount of money is then represented by the number of coins A metal piece (usually a disc) used as money

[For more, click on this link] or banknotes A piece of paper money (especially one issued by a central bank)

[Click link for more facts about this topic] of each denomination. When deciding which denominations to mint (and hence which radices to mix), a compromise is aimed for between a minimal number of different denominations, and a minimal number of individual pieces of coinage required to represent typical quantities.

An example of a mixed radix numeral system A numeral system (or system of numeration) is a framework where a set of numbers are represented by numerals in a consistent manner....

[For more, click on this link] in history is the system of Mayan numerals, which generally used base-20, except for the second place (the "10s" in decimal A number in the decimal system

[For more info, click on this link]) which was base-18, so that the first two places counted up to 360 (an approximation to the number of days in the year).

Mixed-radix numbers of the same base can be manipulated using a generalization of manual arithmetic algorithms. Conversion of values from one mixed base to another is easily accomplished by first converting the place values of the one system into the other, and then applying the digits from the one system against these.

Factorial based radix

Main article: Factoradic In combinatorial mathematics, the factoradic is a specially constructed number....

[For more info, click on this link]

An interesting proposal is a factorial The product of all the integers up to and including a given integer

[For more info, click on this link] based radix, also known as factoradic:

radix: 6 5 4 3 2 1
place value: 5! 4! 3! 2! 1! 0!
decimal: 120 24 6 2 1 1

For example, the biggest number that could be represented with six digits would be 543210 which equals 719 in decimal A number in the decimal system

[Click link for more facts about this topic]: 5×5! + 4×4! + 3×3! + 2×2! + 1×1! + 0×1!. It might not be clear at first sight but factorial based numbering system is also unambiguous. No number can be represented by more than one way because the sum of respective factorials multiplied by the index is always the next factorial minus one:



This can be easily proved with mathematical induction Mathematical induction is a method of mathematical proof typically used to establish that a given statement is true of all natural numbers, or otherwise is true of all members of an infinite sequence....

[For more facts about this topic, click this link].

There is a natural mapping between the integers 0, ..., n! − 1 and permutation Act of changing the lineal order of objects in a group

[Follow this hyperlink for a summary of this subject]s of n elements in lexicographic order, when the integers are expressed in factoradic form.

Answer 5

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Answer 6

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