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Inmathematics apolydivisible number (ormagic number) is anumber in a givennumber base withdigitsabcde... that has the following properties:^[1]

1. Its first digita is not 0.
2. The number formed by its first two digitsab is a multiple of 2.
3. The number formed by its first three digitsabc is a multiple of 3.
4. The number formed by its first four digitsabcd is a multiple of 4.
5. etc.

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

${\displaystyle \lfloor \frac{n}{b^{k-i}}\rfloor \equiv 0(\mathrm{mod}i)}$.

Example

For example, 10801 is a seven-digit polydivisible number inbase 4, as

${\displaystyle \lfloor \frac{10801}{4^{7-1}}\rfloor =\lfloor \frac{10801}{4096}\rfloor =2\equiv 0(\mathrm{mod}1),}$

${\displaystyle \lfloor \frac{10801}{4^{7-2}}\rfloor =\lfloor \frac{10801}{1024}\rfloor =10\equiv 0(\mathrm{mod}2),}$

${\displaystyle \lfloor \frac{10801}{4^{7-3}}\rfloor =\lfloor \frac{10801}{256}\rfloor =42\equiv 0(\mathrm{mod}3),}$

${\displaystyle \lfloor \frac{10801}{4^{7-4}}\rfloor =\lfloor \frac{10801}{64}\rfloor =168\equiv 0(\mathrm{mod}4),}$

${\displaystyle \lfloor \frac{10801}{4^{7-5}}\rfloor =\lfloor \frac{10801}{16}\rfloor =675\equiv 0(\mathrm{mod}5),}$

${\displaystyle \lfloor \frac{10801}{4^{7-6}}\rfloor =\lfloor \frac{10801}{4}\rfloor =2700\equiv 0(\mathrm{mod}6),}$

${\displaystyle \lfloor \frac{10801}{4^{7-7}}\rfloor =\lfloor \frac{10801}{1}\rfloor =10801\equiv 0(\mathrm{mod}7).}$

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

The following table lists maximum polydivisible numbers for some basesb, whereA−Z represent digit values 10 to 35.

Base${\displaystyle b}$	Maximum polydivisible number (OEIS: A109032)	Number of base-b digits (OEIS: A109783)
2	102	2
3	20 02203	6
4	222 03014	7
5	40220 422005	10
10	36085 28850 36840 07860 36725[2][3][4]	25
12	6068 903468 50BA68 00B036 20646412	28

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

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

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

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

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

Base${\displaystyle b}$	${\displaystyle \mathrm{\Sigma }(b)}$	Est. of${\displaystyle \mathrm{\Sigma }(b)}$	Percent Error
2	2	${\displaystyle \frac{e^2-1}{2}\approx 3.1945}$	59.7%
3	15	${\displaystyle \frac{2}{3}(e^3-1)\approx 12.725}$	-15.1%
4	37	${\displaystyle \frac{3}{4}(e^4-1)\approx 40.199}$	8.64%
5	127	${\displaystyle \frac{4}{5}(e^5-1)\approx 117.93}$	−7.14%
10	20456[2]	${\displaystyle \frac{9}{10}(e^{10}-1)\approx 19823}$	-3.09%

All numbers are represented in base${\displaystyle b}$, using A−Z to represent digit values 10 to 35.

Lengthn	F3(n)	Est. of F3(n)	Polydivisible numbers
1	2	2	1, 2
2	3	3	11, 20, 22
3	3	3	110, 200, 220
4	3	2	1100, 2002, 2200
5	2	1	11002, 20022
6	2	1	110020, 200220
7	0	0	${\displaystyle \varnothing }$

Lengthn	F4(n)	Est. of F4(n)	Polydivisible numbers
1	3	3	1, 2, 3
2	6	6	10, 12, 20, 22, 30, 32
3	8	8	102, 120, 123, 201, 222, 300, 303, 321
4	8	8	1020, 1200, 1230, 2010, 2220, 3000, 3030, 3210
5	7	6	10202, 12001, 12303, 20102, 22203, 30002, 32103
6	4	4	120012, 123030, 222030, 321030
7	1	2	2220301
8	0	1	${\displaystyle \varnothing }$

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 withn digits are

1, 11, 110, 1102, 11020, 110204, 1133000, 11330000, 132204314, 1322043140, none...

The largest base 5 polydivisible numbers withn digits are

4, 44, 443, 4431, 44310, 443102, 4431024, 44310242, 443102421, 4022042200, none...

The number of base 5 polydivisible numbers withn digits are

4, 10, 17, 21, 21, 21, 13, 10, 6, 4, 0, 0, 0...

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... (sequenceA144688 in theOEIS)

The smallest base 10 polydivisible numbers withn digits are

1, 10, 102, 1020, 10200, 102000, 1020005, 10200056, 102000564, 1020005640, 10200056405, 102006162060, 1020061620604, 10200616206046, 102006162060465, 1020061620604656, 10200616206046568, 108054801036000018, 1080548010360000180, 10805480103600001800, ... (sequenceA214437 in theOEIS)

The largest base 10 polydivisible numbers withn digits are

9, 98, 987, 9876, 98765, 987654, 9876545, 98765456, 987654564, 9876545640, 98765456405, 987606963096, 9876069630960, 98760696309604, 987606963096045, 9876062430364208, 98485872309636009, 984450645096105672, 9812523240364656789, 96685896604836004260, ... (sequenceA225608 in theOEIS)

The number of base 10 polydivisible numbers withn 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, ... (sequenceA143671 in theOEIS)

The example below searches for polydivisible numbers inPython.

deffind_polydivisible(base:int)->list[int]:"""Find polydivisible number."""numbers=[]previous=[iforiinrange(1,base)]new=[]digits=2whilenotprevious==[]:numbers.append(previous)forninprevious:forjinrange(0,base):number=n*base+jifnumber%digits==0:new.append(number)previous=newnew=[]digits=digits+1returnnumbers

Polydivisible numbers represent a generalization of the following well-known^[2] problem inrecreational 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^[6]

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

• Findingpalindromic 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 apandigital polydivisible number.

• YouTube - a pandigital number that is also polydivisible

• Base-dependent integer sequences
• Modular arithmetic

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