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]

Contents

  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 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 ,

.
Example

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

Enumeration

For any given base , 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.

Base Maximum polydivisible number ( OEIS:  A109032 )Number of base-b digits ( OEIS:  A109783 )
2 1022
3 20 022036
4 222 030147
5 40220 42200510
10 36085 28850 36840 07860 36725 [2] [3] [4] 25
12 6068 903468 50BA68 00B036 2064641228

Estimate for Fb(n) and Σ(b)

Graph of number of
n
{\displaystyle n}
-digit polydivisible numbers in base 10
F
10
(
n
)
{\displaystyle F_{10}(n)}
vs estimate of
F
10
(
n
)
{\displaystyle F_{10}(n)} Graph of polydivisible number vectorial.svg
Graph of number of -digit polydivisible numbers in base 10 vs estimate of

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
2 259.7%
3 15-15.1%
4 378.64%
5 127−7.14%
10 20456 [2] -3.09%

Specific bases

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

Base 2

Length nF2(n)Est. of F2(n)Polydivisible numbers
1111
21110

Base 3

Length nF3(n)Est. of F3(n)Polydivisible numbers
1221, 2
23311, 20, 22
333110, 200, 220
4321100, 2002, 2200
52111002, 20022
621110020, 200220
700

Base 4

Length nF4(n)Est. of F4(n)Polydivisible numbers
1331, 2, 3
26610, 12, 20, 22, 30, 32
388102, 120, 123, 201, 222, 300, 303, 321
4881020, 1200, 1230, 2010, 2220, 3000, 3030, 3210
57610202, 12001, 12303, 20102, 22203, 30002, 32103
644120012, 123030, 222030, 321030
7122220301
801

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...
Length nF5(n)Est. of F5(n)
144
21010
31717
42121
52121
62117
71312
8108
964
1042

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... (sequence A144688 in the OEIS )

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, ... (sequence A214437 in the OEIS )

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, ... (sequence A225608 in the OEIS )

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, ... (sequence A143671 in the OEIS )
Length nF10(n) [5] Est. of F10(n)
199
24545
3150150
4375375
5750750
612001250
717131786
822272232
924922480
1024922480
1122252255
1220411879
1315751445
1411321032
15770688
16571430
17335253
18180141
199074
204437
211817
22128
2363
2431
2511

Programming example

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:

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:

48 000 688 208 466 084 040
30 000 600 003

References

  1. De, Moloy, MATH'S BELIEVE IT OR NOT
  2. 1 2 3 Parker, Matt (2014), "Can you digit?", Things to Make and Do in the Fourth Dimension, Particular Books, pp. 7–8, ISBN   9780374275655 via Google Books
  3. Wells, David (1986), The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, p. 197, ISBN   9780140261493 via Google Books
  4. Lines, Malcolm (1986), "How Do These Series End?", A Number for your Thoughts, Taylor and Francis Group, p. 90, ISBN   9780852744956
  5. (sequence A143671 in the OEIS )
  6. Lanier, Susie, Nine Digit Number