Polydivisible number

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) Its first digit a is not 0.
 * 2) The number formed by its first two digits ab is a multiple of 2.
 * 3) The number formed by its first three digits abc is a multiple of 3.
 * 4) The number formed by its first four digits abcd is a multiple of 4.
 * 5) etc.

Definition
Let $$n$$ be a positive integer, and let $$k = \lfloor \log_{b}{n} \rfloor + 1$$ be the number of digits in n written in base b. The number n is a polydivisible number if for all $$1 \leq i \leq k$$,
 * $$\left\lfloor\frac{n}{b^{k - i}}\right\rfloor \equiv 0 \pmod i$$.


 * Example

For example, 10801 is a seven-digit polydivisible number in base 4, as
 * $$\left\lfloor\frac{10801}{4^{7 - 1}}\right\rfloor = \left\lfloor\frac{10801}{4096}\right\rfloor = 2 \equiv 0 \pmod 1,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 2}}\right\rfloor = \left\lfloor\frac{10801}{1024}\right\rfloor = 10 \equiv 0 \pmod 2,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 3}}\right\rfloor = \left\lfloor\frac{10801}{256}\right\rfloor = 42 \equiv 0 \pmod 3,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 4}}\right\rfloor = \left\lfloor\frac{10801}{64}\right\rfloor = 168 \equiv 0 \pmod 4,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 5}}\right\rfloor = \left\lfloor\frac{10801}{16}\right\rfloor = 675 \equiv 0 \pmod 5,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 6}}\right\rfloor = \left\lfloor\frac{10801}{4}\right\rfloor = 2700 \equiv 0 \pmod 6,$$
 * $$\left\lfloor\frac{10801}{4^{7 - 7}}\right\rfloor = \left\lfloor\frac{10801}{1}\right\rfloor = 10801 \equiv 0 \pmod 7.$$

Enumeration
For any given base $$b$$, there are only a finite number of polydivisible numbers.

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

Estimate for Fb(n) and &Sigma;(b)


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

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


 * $$F_b(n) \approx (b - 1)\frac{b^{n-1}}{n!}.$$

Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately


 * $$\Sigma(b) \approx \frac{b - 1}{b}(e^{b}-1)$$

Specific bases
All numbers are represented in base $$b$$, using A−Z to represent digit values 10 to 35.

Base 5
The polydivisible numbers in base 5 are
 * 1, 2, 3, 4, 11, 13, 20, 22, 24, 31, 33, 40, 42, 44, 110, 113, 132, 201, 204, 220, 223, 242, 311, 314, 330, 333, 402, 421, 424, 440, 443, 1102, 1133, 1322, 2011, 2042, 2200, 2204, 2231, 2420, 2424, 3113, 3140, 3144, 3302, 3333, 4022, 4211, 4242, 4400, 4404, 4431, 11020, 11330, 13220, 20110, 20420, 22000, 22040, 22310, 24200, 24240, 31130, 31400, 31440, 33020, 33330, 40220, 42110, 42420, 44000, 44040, 44310, 110204, 113300, 132204, 201102, 204204, 220000, 220402, 223102, 242000, 242402, 311300, 314000, 314402, 330204, 333300, 402204, 421102, 424204, 440000, 440402, 443102, 1133000, 1322043, 2011021, 2042040, 2204020, 2420003, 2424024, 3113002, 3140000, 3144021, 4022042, 4211020, 4431024, 11330000, 13220431, 20110211, 20420404, 24200031, 31400004, 31440211, 40220422, 42110202, 44310242, 132204314, 201102110, 242000311, 314000044, 402204220, 443102421, 1322043140, 2011021100, 3140000440, 4022042200

The smallest base 5 polydivisible numbers with n digits are
 * 1, 11, 110, 1102, 11020, 110204, 1133000, 11330000, 132204314, 1322043140, none...

The largest base 5 polydivisible numbers with n digits are
 * 4, 44, 443, 4431, 44310, 443102, 4431024, 44310242, 443102421, 4022042200, none...

The number of base 5 polydivisible numbers with n digits are
 * 4, 10, 17, 21, 21, 21, 13, 10, 6, 4, 0, 0, 0...

Base 10
The polydivisible numbers in base 10 are
 * 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, 46, 48, 50, 52, 54, 56, 58, 60, 62, 64, 66, 68, 70, 72, 74, 76, 78, 80, 82, 84, 86, 88, 90, 92, 94, 96, 98, 102, 105, 108, 120, 123, 126, 129, 141, 144, 147, 162, 165, 168, 180, 183, 186, 189, 201, 204, 207, 222, 225, 228, 243, 246, 249, 261, 264, 267, 282, 285, 288...

The smallest base 10 polydivisible numbers with n digits are
 * 1, 10, 102, 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640, 10200056405, 102006162060, 1020061620604, 10200616206046, 102006162060465, 1020061620604656, 10200616206046568, 108054801036000018, 1080548010360000180, 10805480103600001800, ...

The largest base 10 polydivisible numbers with n digits are
 * 9, 98, 987, 9876, 98765, 987654, 9876545, 98765456, 987654564, 9876545640, 98765456405, 987606963096, 9876069630960, 98760696309604, 987606963096045, 9876062430364208, 98485872309636009, 984450645096105672, 9812523240364656789, 96685896604836004260, ...

The number of base 10 polydivisible numbers with n digits are
 * 9, 45, 150, 375, 750, 1200, 1713, 2227, 2492, 2492, 2225, 2041, 1575, 1132, 770, 571, 335, 180, 90, 44, 18, 12, 6, 3, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...

Programming example
The example below searches for polydivisible numbers in Python.

Related problems
Polydivisible numbers represent a generalization of the following well-known problem in recreational mathematics:


 * Arrange the digits 1 to 9 in order so that the first two digits form a multiple of 2, the first three digits form a multiple of 3, the first four digits form a multiple of 4 etc. and finally the entire number is a multiple of 9.

The solution to the problem is a nine-digit polydivisible number with the additional condition that it contains the digits 1 to 9 exactly once each. There are 2,492 nine-digit polydivisible numbers, but the only one that satisfies the additional condition is


 * 381 654 729

Other problems involving polydivisible numbers include:


 * Finding polydivisible numbers with additional restrictions on the digits - for example, the longest polydivisible number that only uses even digits is


 * 48 000 688 208 466 084 040


 * Finding palindromic polydivisible numbers - for example, the longest palindromic polydivisible number is


 * 30 000 600 003


 * A common, trivial extension of the aforementioned example is to arrange the digits 0 to 9 to make a 10 digit number in the same way, the result is 3816547290. This is a pandigital polydivisible number.