scholarly journals Divisibility of Fibonomial coefficients in terms of their digital representations and applications

2022 ◽  
Vol 7 (4) ◽  
pp. 5314-5327
Author(s):  
Phakhinkon Napp Phunphayap ◽  
◽  
Prapanpong Pongsriiam
Keyword(s):  
Asymptotic Formulas ◽  
Digital Representations ◽  
Fibonomial Coefficients ◽  
Fibonomial Coefficient ◽  
Positive Integers

<abstract><p>We give a characterization for the integers $ n \geq 1 $ such that the Fibonomial coefficient $ {pn \choose n}_F $ is divisible by $ p $ for any prime $ p \neq 2, 5 $. Then we use it to calculate asymptotic formulas for the number of positive integers $ n \leq x $ such that $ p \mid {pn \choose n}_F $. This completes the study on this problem for all primes $ p $.</p></abstract>

scholarly journals ON THE AVERAGE NUMBER OF SUBGROUPS OF THE GROUP ℤm × ℤn

2014 ◽  
Vol 10 (02) ◽  
pp. 363-374 ◽  
Author(s):  
WERNER GEORG NOWAK ◽  
LÁSZLÓ TÓTH
Keyword(s):  
Asymptotic Formulas ◽  
Residue Classes ◽  
Positive Integers

Let ℤm be the group of residue classes modulo m. Let s(m, n) denote the total number of subgroups of the group ℤm × ℤn, where m and n are arbitrary positive integers. We derive asymptotic formulas for the sum ∑m,n≤x s(m, n) and for the corresponding sum restricted to gcd (m, n) > 1 which concerns the groups ℤm × ℤn having rank two.

scholarly journals On Certain Sums of Arithmetic Functions Involving the GCD and LCM of Two Positive Integers

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Randell Heyman ◽  
László Tóth
Keyword(s):  
Arithmetic Functions ◽  
Asymptotic Formulas ◽  
Common Generalization ◽  
Positive Integers

AbstractWe obtain asymptotic formulas with remainder terms for the hyperbolic summations $$\sum _{mn\le x} f((m,n))$$ ∑ m n ≤ x f ( ( m , n ) ) and $$\sum _{mn\le x} f([m,n])$$ ∑ m n ≤ x f ( [ m , n ] ) , where f belongs to certain classes of arithmetic functions, (m, n) and [m, n] denoting the gcd and lcm of the integers m, n. In particular, we investigate the functions $$f(n)=\tau (n), \log n, \omega (n)$$ f ( n ) = τ ( n ) , log n , ω ( n ) and $$\Omega (n)$$ Ω ( n ) . We also define a common generalization of the latter three functions, and prove a corresponding result.

scholarly journals A Note on the Asymptotic Behavior of the Distribution Function of a General Sequence

2021 ◽  
Vol 35 (1) ◽  
pp. 44-54
Author(s):  
Reza Farhadian ◽  
Rafael Jakimczuk
Keyword(s):  
Distribution Function ◽  
Asymptotic Behavior ◽  
Asymptotic Formulas ◽  
Bell Numbers ◽  
General Sequence ◽  
Positive Integers

Abstract The aim of this note is to study the distribution function of certain sequences of positive integers, including, for example, Bell numbers, factorials and primorials. In fact, we establish some general asymptotic formulas in this regard. We also prove some limits that connect these sequences with the number e. Furthermore, we present a generalization of the primorial.

scholarly journals Sums of weighted averages of gcd-sum functions

2018 ◽  
Vol 14 (10) ◽  
pp. 2699-2728 ◽  
Author(s):  
Isao Kiuchi ◽  
Sumaia Saad eddin
Keyword(s):  
Mean Value ◽  
Arithmetical Function ◽  
Partial Sums ◽  
Asymptotic Formulas ◽  
Greatest Common Divisor ◽  
Fixed Integer ◽  
Weighted Averages ◽  
Error Terms ◽  
Positive Integers

Let [Formula: see text] be the greatest common divisor of the integers [Formula: see text] and [Formula: see text]. In this paper, we give several interesting asymptotic formulas for weighted averages of the [Formula: see text]-sum function [Formula: see text] and the function [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text], namely [Formula: see text] with any fixed integer [Formula: see text] and any arithmetical function [Formula: see text]. We also establish mean value formulas for the error terms of asymptotic formulas for partial sums of [Formula: see text]-sum functions [Formula: see text]

Language, procedures, and the non-perceptual origin of natural number concepts

2016 ◽  
Author(s):  
David Barner
Keyword(s):  
General Framework ◽  
Ad Hoc ◽  
Building Blocks ◽  
Number Words ◽  
Linguistic Markers ◽  
Logical Foundations ◽  
Large Numbers ◽  
Perceptual Representations ◽  
Positive Integers ◽  
Perceptual Systems

Perceptual representations – e.g., of objects or approximate magnitudes –are often invoked as building blocks that children combine with linguisticsymbols when they acquire the positive integers. Systems of numericalperception are either assumed to contain the logical foundations ofarithmetic innately, or to supply the basis for their induction. Here Ipropose an alternative to this general framework, and argue that theintegers are not learned from perceptual systems, but instead arise toexplain perception as part of language acquisition. Drawing oncross-linguistic data and developmental data, I show that small numbers(1-4) and large numbers (~5+) arise both historically and in individualchildren via entirely distinct mechanisms, constituting independentlearning problems, neither of which begins with perceptual building blocks.Specifically, I propose that children begin by learning small numbers(i.e., *one, two, three*) using the same logical resources that supportother linguistic markers of number (e.g., singular, plural). Several yearslater, children discover the logic of counting by inferring the logicalrelations between larger number words from their roles in blind countingprocedures, and only incidentally associate number words with perception ofapproximate magnitudes, in an *ad hoc* and highly malleable fashion.Counting provides a form of explanation for perception but is not causallyderived from perceptual systems.

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

On a Two-Sided Turán Problem

10.37236/1735 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Dhruv Mubayi ◽  
Yi Zhao
Keyword(s):  
Minimum Size ◽  
Turan Problem ◽  
Turan Problems ◽  
Turán Problem ◽  
Positive Integers

Given positive integers $n,k,t$, with $2 \le k\le n$, and $t < 2^k$, let $m(n,k,t)$ be the minimum size of a family ${\cal F}$ of nonempty subsets of $[n]$ such that every $k$-set in $[n]$ contains at least $t$ sets from ${\cal F}$, and every $(k-1)$-set in $[n]$ contains at most $t-1$ sets from ${\cal F}$. Sloan et al. determined $m(n, 3, 2)$ and Füredi et al. studied $m(n, 4, t)$ for $t=2, 3$. We consider $m(n, 3, t)$ and $m(n, 4, t)$ for all the remaining values of $t$ and obtain their exact values except for $k=4$ and $t= 6, 7, 11, 12$. For example, we prove that $ m(n, 4, 5) = {n \choose 2}-17$ for $n\ge 160$. The values of $m(n, 4, t)$ for $t=7,11,12$ are determined in terms of well-known (and open) Turán problems for graphs and hypergraphs. We also obtain bounds of $m(n, 4, 6)$ that differ by absolute constants.

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