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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 ,
For example, 10801 is a seven-digit polydivisible number in base 4, as
For any given base , there are only a finite number of polydivisible numbers.
The following table lists maximum polydivisible numbers for some bases b, where A−Z represent digit values 10 to 35.
Base | 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 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 .
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 :
Summing over all values of n, this estimate suggests that the total number of polydivisible numbers will be approximately
Base | Est. of | Percent Error | |
---|---|---|---|
2 | 2 | 59.7% | |
3 | 15 | -15.1% | |
4 | 37 | 8.64% | |
5 | 127 | −7.14% | |
10 | 20456 [2] | -3.09% | |
All numbers are represented in base , using A−Z to represent digit values 10 to 35.
Length n | F2(n) | Est. of F2(n) | Polydivisible numbers |
---|---|---|---|
1 | 1 | 1 | 1 |
2 | 1 | 1 | 10 |
Length n | 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 | |
Length n | 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 | |
The polydivisible numbers in base 5 are
The smallest base 5 polydivisible numbers with n digits are
The largest base 5 polydivisible numbers with n digits are
The number of base 5 polydivisible numbers with n digits are
Length n | F5(n) | Est. of F5(n) |
---|---|---|
1 | 4 | 4 |
2 | 10 | 10 |
3 | 17 | 17 |
4 | 21 | 21 |
5 | 21 | 21 |
6 | 21 | 17 |
7 | 13 | 12 |
8 | 10 | 8 |
9 | 6 | 4 |
10 | 4 | 2 |
The polydivisible numbers in base 10 are
The smallest base 10 polydivisible numbers with n digits are
The largest base 10 polydivisible numbers with n digits are
The number of base 10 polydivisible numbers with n digits are
Length n | F10(n) [5] | Est. of F10(n) |
---|---|---|
1 | 9 | 9 |
2 | 45 | 45 |
3 | 150 | 150 |
4 | 375 | 375 |
5 | 750 | 750 |
6 | 1200 | 1250 |
7 | 1713 | 1786 |
8 | 2227 | 2232 |
9 | 2492 | 2480 |
10 | 2492 | 2480 |
11 | 2225 | 2255 |
12 | 2041 | 1879 |
13 | 1575 | 1445 |
14 | 1132 | 1032 |
15 | 770 | 688 |
16 | 571 | 430 |
17 | 335 | 253 |
18 | 180 | 141 |
19 | 90 | 74 |
20 | 44 | 37 |
21 | 18 | 17 |
22 | 12 | 8 |
23 | 6 | 3 |
24 | 3 | 1 |
25 | 1 | 1 |
The example below searches for polydivisible numbers in Python.
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 in recreational mathematics:
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
Other problems involving polydivisible numbers include: