On the growth and bounds for a certain sequence of integers known as Bogotá numbers

Question

A Bogotá number is a non-negative 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).

Bogotá numbers up to 1000 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, 795, 819, 848, 884, 900, 912, 933, 944, 966, 992.

It has been shown that the natural density of Bogotá numbers is 0: https://math.stackexchange.com/questions/3713294/on-the-density-of-a-certain-sequence-of-integers.

The number of Bogotá numbers, B (n), less than or equal to n, for n = 10^ 0, 1, 2, 3..., 9 is 2, 4, 19, 67, 280, 1166, 4777, 19899, 82278, and 340649 as calculated by Freddy Barrera.

Crude estimates and bounds for the value of B (n) are not difficult to come by. How precise can we get?

Answer 1

Fix an integer $b > 1$ and let $p_b(n)$ denote the product of the base-$b$ digits of the integer $n$. Then let $\mathcal{B}_b$ be the set of numbers of the form $p_b(n)n$, for some integer $n \geq 0$, and put $\mathcal{B}_b(x) := \mathcal{B}_b \cap [0, x]$ for all $x > 0$.

The OP is asking for bounds for $\mathcal{B}_b(x)$ (in the special case $b=10$).

I guess that using the same methods of [1], one should be able to prove an upper of the form $x^{c_b + o(1)}$ for some constant $c_b$ depending on $b$. The details are many so I refer directly to the paper (the version on arXiv):

• For a parameter $\alpha > 0$, one splits $\mathcal{B}_b$ into two subsets: $\mathcal{B}_b^\prime$, consisting of the numbers with $p_b(n) > x^\alpha$, and $\mathcal{B}_b^{\prime\prime}$, the remaining numbers with $p_b(n) \leq x^\alpha$.

• An upper bound for $\mathcal{B}_b^\prime(x)$ is given considering that each of its elements has a $b$-smooth divisor $> x^\alpha$ (pag. 3, (6) and the three equations before)

• An upper bound for $\mathcal{B}_b^{\prime\prime}(x)$ is given by estimating a sum of multinomial coefficients (pag. 4-5, (11) and the two equations before).

[1] C. Sanna, On numbers divisible by the product of their nonzero base b digits, Quaestiones Mathematicae (in press) https://arxiv.org/abs/1809.05463