# The Number of Integers Divisible by a^k Except a^(k+l) b

## Authors

• Yanapat Tongron Mathematics and Applied Statistics Program, Faculty of Science and Technology, Nakhon Ratchasima Rajabhat University, Nakhon Ratchasima 30000, Thailand
• Kanyaphak Paikhlaew Mathematics and Applied Statistics Program, Faculty of Science and Technology, Nakhon Ratchasima Rajabhat University, Nakhon Ratchasima 30000, Thailand

## Keywords:

Bounds, Ceiling function, Divisibility, Floor function, Integer

## Abstract

For positive integers $a,b,k,l,n$, let $A_{(k,l)}^{(a,b)}(n)$ be the set of all integers divisible by $a^{k}$ except $a^{k+l}b$ between $1$ and $n$. Under certain condition on these integers, we obtain the explicit formula for $\left&space;|A_{(k,l)}^{(a,b)}(n)&space;\right&space;|$, where $\left&space;|X&space;\right&space;|$ denotes the number of elements in a finite set $X$. We also provide bounds on $\left&space;|A_{(k,l)}^{(a,b)}(n)&space;\right&space;|$ and show that these are the best bounds by examples.

2022-09-28

## How to Cite

Tongron , Y., & Paikhlaew, K. (2022). The Number of Integers Divisible by a^k Except a^(k+l) b. Science & Technology Asia, 27(3), 132–136. Retrieved from https://ph02.tci-thaijo.org/index.php/SciTechAsia/article/view/242455

## Section

Physical sciences