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

Yanapat Tongron; Kanyaphak Paikhlaew

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

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Published

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