Polydivisible number

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In mathematics a polydivisible number (or magic number) is a number in a given number base with digits abcde... that has the following properties:^[1]

Let be a positive integer, and let be the number of digits in n written in base b. The number n is a polydivisible number if for all ,${\displaystyle n}$${\displaystyle k=\lfloor \log _{b}{n}\rfloor +1}$${\displaystyle 1\leq i\leq k}$

For any given base , there are only a finite number of polydivisible numbers.${\displaystyle b}$

The following table lists maximum polydivisible numbers for some bases b, where A−Z represent digit values 10 to 35.

Let be the number of digits. The function determines the number of polydivisible numbers that has digits in base , and the function is the total number of polydivisible numbers in base .${\displaystyle n}$${\displaystyle F_{b}(n)}$${\displaystyle n}$${\displaystyle b}$${\displaystyle \Sigma (b)}$${\displaystyle b}$

If is a polydivisible number in base with digits, then it can be extended to create a polydivisible number with digits if there is a number between and that is divisible by . If is less or equal to , then it is always possible to extend an digit polydivisible number to an -digit polydivisible number in this way, and indeed there may be more than one possible extension. If is greater than , it is not always possible to extend a polydivisible number in this way, and as becomes larger, the chances of being able to extend a given polydivisible number become smaller. On average, each polydivisible number with digits can be extended to a polydivisible number with digits in different ways. This leads to the following estimate for :${\displaystyle k}$${\displaystyle b}$${\displaystyle n-1}$${\displaystyle n}$${\displaystyle bk}$${\displaystyle b(k+1)-1}$${\displaystyle n}$${\displaystyle n}$${\displaystyle b}$${\displaystyle n-1}$${\displaystyle n}$${\displaystyle n}$${\displaystyle b}$${\displaystyle n}$${\displaystyle n-1}$${\displaystyle n}$${\displaystyle {\frac {b}{n}}}$${\displaystyle F_{b}(n)}$

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