Bogotá Numbers

Yesterday I turned 26550 days old and one of the properties of 26550 is that it's a Bogotá number. These numbers comprise OEIS A336826:

A336826

Bogotá numbers: numbers k such that k = m*p(m) where p(m) is the digital product of m.

As an example, \(26550 = 295 \times (2 \times 9 \times 5)\). It's interesting to note that the digital product of any number is of the form \(2^a \times 3^b \times 5^c \times 7^c\) where \(a,b,c,d\) are non-negative integers (and thus could include zero values). Numbers of this form are said to be 7-smooth and in general an \(n\)-smooth number is defined as one whose prime factors are all less than or equal to \(n\). In the case of 295 it can be seen that the digital product is of the form \(2^1 \times 3^2 \times 5^1 \times 7^0\).

The initial Bogotá numbers are:

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

In the above list the pairs of consecutive Bogotá numbers are marked in bold. These pairs are not that frequent. Here are the initial pairs (permalink):

  First Member   Second Member

  0              1
  24             25
  2510           2511
  5210           5211
  8991           8992
  56384          56385
  348732         348733
  460719         460720
  867839         867840
  28997919       28997920
  254181375      254181376

Up to one million, the record gaps between successive Bogotá numbers are (permalink):

Number         Next number    Gap

  0              1              1
  1              4              3
  4              9              5
  16             24             8
  25             36             11
  96             111            15
  119            138            19
  144            164            20
  171            192            21
  192            224            32
  255            297            42
  575            648            73
  1778           1872           94
  2688           2784           96
  3942           4092           150
  6125           6288           163
  6792           6966           174
  9144           9333           189
  9468           9666           198
  10820          11034          214
  14256          14488          232
  16119          16408          289
  17295          17676          381
  43040          43512          472
  84996          85608          612
  90272          91152          880
  185616         186672         1056
  213346         214416         1070
  285408         286920         1512
  334950         336672         1722
  853056         855036         1980
  949176         951264         2088

The natural density of Bogotá numbers is 0 (link) and the numbers less than or equal to \(10^n\) with \(n=0,1, 2, \dots, 9\) are 2, 4, 19, 67, 280, 1166, 4777, 19899, 82278, and 340649 respectively (link).

Bogotá of course is the capital of Columbia and these numbers were so named by Tomás Uribe and Juan Pablo Fernández based on similarity to the construction of the Colombian numbers or self numbers. These latter numbers are ones that cannot be formed from \(m\) + digit sum of \(m\) for some positive integer \(m\). I've written about these in a post on October 25th 2018 titled Self Numbers and Junction Numbers.

OEIS A336984 lists numbers that are both Columbian and Bogotá numbers:

A336984

Colombian numbers that are also Bogotá numbers.

The initial members of this sequence are:

1, 9, 42, 64, 75, 255, 312, 378, 525, 648, 738, 1111, 1278, 2224, 2448, 2784, 2817, 3504, 3864, 3875, 4977, 5238, 5495, 5888, 8992, 9712, 10368, 11358, 11817, 12348, 12875, 13136, 13584, 13775, 13832, 13944, 15351, 15384, 15744, 15900, 16912, 17768, 18095, 19344, 20448