# Pairs of Bogotá Numbers

A Bogotá number is a positive integer equal to some smaller number, or itself, times its digital product, i.e. the product of its digits. For example, 138 is a Bogotá number because 138 = 23 x (2 x 3).

24 and 25 are the first instance of two consecutive numbers both of which are Bogotá numbers. Indeed 24 = 12 x (1 x 2), while 25 = 5 x (5).

i) Find the next five pairs of consecutive numbers consisting of two Bogotá numbers.

ii) Are there infinitely many such pairs?

iii) Can arbitrarily long sets of consecutive numbers be found all of which are Bogotá numbers?

https://oeis.org/A336826

• I dont understand, the product of the digits of 138 is 24 not 23. Same for 24: the product is 8, for 25 the product is 10. What do I miss??
– daw
Jun 9 '20 at 14:09
• @daw If I understand correctly, "It" in the "its digital product" is referring to the smaller number, not the Bogota number. So 23 times (the digital product of 23) is 138. Jun 9 '20 at 14:23
• @bobble No, it isn´t. But as Picasso said: “Computers are useless. They can only give you answers”. Jun 9 '20 at 15:18
• The first 4 pairs are $(2510,2511), (5210, 5211), (8991, 8992), (56384, 56385)$ Jun 9 '20 at 15:24
• For clarity, we can define a Bogotá number as a number $m$ for which there exists a number $n$ such that $m=n \times p(n)$, where $p(n)$ is the digital product of $n$. Jun 9 '20 at 17:27

I'm going to call the number that generates a Bogotá number a Bogotá root.

i) Find the next five pairs of consecutive numbers consisting of two Bogotá numbers.


The first eight pairs are:

24 <- 12
25 <- 5

2510 <- 251
2511 <- 93

5210 <- 521
5211 <- 193

8991 <- 333
8992 <- 1124

56384 <- 881
56385 <- 537

348732 <- 3229
348733 <- 7117

460719 <- 7313
460720 <- 11518

867839 <- 17711
867840 <- 5424

Two additional pairs with Bogotá roots less than 1,000,000 are:

28997919 <- 119333
28997920 <- 51782

254181375 <- 53795
254181376 <- 248224

ii) Are there infinitely many such pairs?


Possible, but they seem fairly sparse.

iii) Can arbitrarily long sets of consecutive numbers be found all of which are Bogotá numbers?


The only sequence of more than two I can suggest twists the definition a bit:

-1 <- -1
0 <- 0 or 10 or 5103 or ...
1 <- 1

## Other observations

There are a number of Bogotá numbers with multiple Bogotá roots. There are 3905 numbers with multiple Bogotá roots where the roots are under 1000000. The first 10 are:

192 <- 24 32
648 <- 36 81
819 <- 91 117
1197 <- 133 171
1536 <- 48 64
4872 <- 87 174
4977 <- 79 711
5976 <- 166 332
7056 <- 98 441
9968 <- 178 712

and a few more with more Bogotá roots:

549504 <- 1696 2862 3392 3816
1798848 <- 6246 12492 33312
4193856 <- 19416 21843 29124
4804128 <- 4766 16681 21447
5827584 <- 8672 17344 182112
7426944 <- 7368 12894 14736
1578092544 <- 86976 97848 342468 913248

Some patterns of Bogotá numbers:

No Bogotá root contains a 0. These all generate 0, which violates the definition.

Any number composed of only the digit 1 is a Bogotá number. These are also their own Bogotá root.

Any number composed of any number of the digit 2 and one digit 4 is a Bogotá numbers. Similarly, for the digits 3 and 9, and for the digit 4 and digit sequence 56. Thus, the following are Bogotá numbers: 4, 9, 56, 222222422, 93333333333, 445644444444.

There are similar patterns for Bogotá roots composed of all but one digits being 1.

Edit:

I did some work generating odd Bogotá numbers trying to find consecutive odd Bogotá numbers. Based on the definition, all the digits in the root must be odd, which made the search space somewhat smaller. I found three: (9,11), (8197,8199), and (11977,11979). None of these is part of a consecutive pair. This was for Bogotá roots up to 10 billion, except for 56 that overflowed a 64 bit integer.

Since getting 4 or more consecutive Bogotá numbers requires two consecutive odd ones, I think it unlikely to find longer runs.

• math.stackexchange.com/questions/3713294/… Jun 10 '20 at 0:15
• Here is the list of Bogotá numbers not greater than 1000: 0, 1, 4, 9, 11, 16, 24, 25, 36, 39, 42, 49, 56, 64, 75, 81, 88, 93, 96, 111, 119, 138, 144, 164, 171, 192, 224, 242, 250, 255, 297, 312, 336, 339, 366, 378, 393, 408, 422, 448, 456, 488, 497, 516, 520, 522, 525, 564, 575, 648, 696, 704, 738, 744, 755, 777, 792, 795, 819, 848, 884, 900, 912, 933, 944, 966, 992 Jun 10 '20 at 14:51