On new arithmetic function relative to a fixed positive integer. Part 1

Let n be an arbitrary but fixed positive integer. Let Tn be the set of all monotone - increasing n-tuples of positive integers:1Define2In this note we prove that ϕ is a 1–1 mapping from Tn onto {1, 2, 3,…}.

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Hyperstability of the k -Cubic Functional Equation in Non-Archimedean Banach Spaces

Journal of Mathematics ◽

10.1155/2020/8843464 ◽

2020 ◽

Vol 2020 ◽

pp. 1-10

Author(s):

Youssef Aribou ◽

Mohamed Rossafi

Keyword(s):

Functional Equation ◽

Fixed Point ◽

Positive Integer ◽

Banach Spaces ◽

Functional Equations ◽

Cubic Functional Equation ◽

Fixed Point Approach ◽

Fixed Positive Integer

Using the fixed point approach, we investigate a general hyperstability results for the following k -cubic functional equations f k x + y + f k x − y = k f x + y + k f x − y + 2 k k 2 − 1 f x , where k is a fixed positive integer ≥ 2 , in ultrametric Banach spaces.

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Further Results on a Curious Arithmetic Function

Journal of Mathematics ◽

10.1155/2020/1894162 ◽

2020 ◽

Vol 2020 ◽

pp. 1-7

Author(s):

Long Chen ◽

Kaimin Cheng ◽

Tingting Wang

Keyword(s):

Positive Integer ◽

Prime Number ◽

Arithmetic Function ◽

Rational Numbers ◽

Arithmetic Properties ◽

Adic Valuation ◽

Positive Integers

Let p be an odd prime number and n be a positive integer. Let vpn, N∗, and Q+ denote the p-adic valuation of the integer n, the set of positive integers, and the set of positive rational numbers, respectively. In this paper, we introduce an arithmetic function fp:N∗⟶Q+ defined by fpn≔n/pvpn1−vpn for any positive integer n. We show several interesting arithmetic properties about that function and then use them to establish some curious results involving the p-adic valuation. Some of these results extend Farhi’s results from the case of even prime to that of odd prime.

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On the sum of digits of some sequences of integers

Open Mathematics ◽

10.2478/s11533-012-0080-0 ◽

2013 ◽

Vol 11 (1) ◽

Author(s):

Javier Cilleruelo ◽

Florian Luca ◽

Juanjo Rué ◽

Ana Zumalacárregui

Keyword(s):

Number Theory ◽

Positive Integer ◽

Integer Sequences ◽

Sum Of Digits ◽

Fixed Positive Integer ◽

Almost All

AbstractLet b ≥ 2 be a fixed positive integer. We show for a wide variety of sequences {a n}n=1∞ that for almost all n the sum of digits of a n in base b is at least c b log n, where c b is a constant depending on b and on the sequence. Our approach covers several integer sequences arising from number theory and combinatorics.

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Upper and Lower Bounds for a Function Related to Brown's Lemma

Integers ◽

10.1515/integ.2010.051 ◽

2010 ◽

Vol 10 (6) ◽

Author(s):

Hayri Ardal

Keyword(s):

Positive Integer ◽

Lower Bounds ◽

Upper And Lower Bounds ◽

Fixed Positive Integer ◽

Positive Integers

AbstractThe well-known Brown's lemma says that for every finite coloring of the positive integers, there exist a fixed positive integer

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A generalization of the practical numbers

International Journal of Number Theory ◽

10.1142/s1793042118500902 ◽

2018 ◽

Vol 14 (05) ◽

pp. 1487-1503

Author(s):

Nicholas Schwab ◽

Lola Thompson

Keyword(s):

Positive Integer ◽

Arithmetic Function ◽

Arithmetic Functions ◽

Asymptotic Density ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Necessary And Sufficient ◽

First Time

A positive integer [Formula: see text] is practical if every [Formula: see text] can be written as a sum of distinct divisors of [Formula: see text]. One can generalize the concept of practical numbers by applying an arithmetic function [Formula: see text] to each of the divisors of [Formula: see text] and asking whether all integers in a certain interval can be expressed as sums of [Formula: see text]’s, where the [Formula: see text]’s are distinct divisors of [Formula: see text]. We will refer to such [Formula: see text] as “[Formula: see text]-practical”. In this paper, we introduce the [Formula: see text]-practical numbers for the first time. We give criteria for when all [Formula: see text]-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct [Formula: see text]-practical sets with any asymptotic density, and prove a series of results related to the distribution of [Formula: see text]-practical numbers for many well-known arithmetic functions [Formula: see text].

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On the Stability of Quadratic Functional Equations

Abstract and Applied Analysis ◽

10.1155/2008/628178 ◽

2008 ◽

Vol 2008 ◽

pp. 1-8 ◽

Cited By ~ 9

Author(s):

Jung Rye Lee ◽

Jong Su An ◽

Choonkil Park

Keyword(s):

Functional Equation ◽

Positive Integer ◽

Banach Spaces ◽

Functional Equations ◽

Vector Spaces ◽

Quadratic Functional ◽

The Stability ◽

Fixed Positive Integer ◽

Rassias Stability

LetX,Ybe vector spaces andka fixed positive integer. It is shown that a mappingf(kx+y)+f(kx-y)=2k2f(x)+2f(y)for allx,y∈Xif and only if the mappingf:X→Ysatisfiesf(x+y)+f(x-y)=2f(x)+2f(y)for allx,y∈X. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

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On the generalized Ramanujan-Nagell equation $ x^2+(2k-1)^y = k^z $ with $ k\equiv 3 $ (mod 4)

AIMS Mathematics ◽

10.3934/math.2021615 ◽

2021 ◽

Vol 6 (10) ◽

pp. 10596-10601

Author(s):

Yahui Yu ◽

◽

Jiayuan Hu ◽

Keyword(s):

Positive Integer ◽

Unsolved Problem ◽

Integer Solution ◽

Positive Integer Solution ◽

Elementary Methods ◽

Fixed Positive Integer ◽

Almost All ◽

Terai’S Conjecture ◽

Positive Integers

<abstract><p>Let $ k $ be a fixed positive integer with $ k > 1 $. In 2014, N. Terai <sup>[<xref ref-type="bibr" rid="b6">6</xref>]</sup> conjectured that the equation $ x^2+(2k-1)^y = k^z $ has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. This is still an unsolved problem as yet. For any positive integer $ n $, let $ Q(n) $ denote the squarefree part of $ n $. In this paper, using some elementary methods, we prove that if $ k\equiv 3 $ (mod 4) and $ Q(k-1)\ge 2.11 $ log $ k $, then the equation has only the positive integer solution $ (x, y, z) = (k-1, 1, 2) $. It can thus be seen that Terai's conjecture is true for almost all positive integers $ k $ with $ k\equiv 3 $(mod 4).</p></abstract>

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An optimal algorithm to find minimum k-hop dominating set of interval graphs

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830919500162 ◽

2019 ◽

Vol 11 (02) ◽

pp. 1950016 ◽

Cited By ~ 2

Author(s):

Sambhu Charan Barman ◽

Madhumangal Pal ◽

Sukumar Mondal

Keyword(s):

Positive Integer ◽

Optimal Algorithm ◽

Dominating Set ◽

Interval Graphs ◽

Fixed Positive Integer

For a fixed positive integer [Formula: see text], a [Formula: see text]-hop dominating set [Formula: see text] of a graph [Formula: see text] is a subset of [Formula: see text] such that every vertex [Formula: see text] is within [Formula: see text]-steps from at least one vertex [Formula: see text], i.e., [Formula: see text]. A [Formula: see text]-hop dominating set [Formula: see text] is said to be minimal if there does not exist any [Formula: see text] such that [Formula: see text] is a [Formula: see text]-hop dominating set of G. A dominating set [Formula: see text] is said to be minimum [Formula: see text]-hop dominating set, if it is minimal as well as it is [Formula: see text]-hop dominating set. In this paper, we present an optimal algorithm to find a minimum [Formula: see text]-hop dominating set of interval graphs with [Formula: see text] vertices which runs in [Formula: see text] time.

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On the Growth of Some Functions Related to z(n)

Mathematics ◽

10.3390/math8060876 ◽

2020 ◽

Vol 8 (6) ◽

pp. 876 ◽

Cited By ~ 1

Author(s):

Pavel Trojovský

Keyword(s):

Positive Integer ◽

Arithmetic Function ◽

Fibonacci Sequence ◽

Upper Bounds ◽

Integer Solution ◽

Lower And Upper Bounds ◽

Positive Integer Solution

The order of appearance z : Z > 0 → Z > 0 is an arithmetic function related to the Fibonacci sequence ( F n ) n . This function is defined as the smallest positive integer solution of the congruence F k ≡ 0 ( mod n ) . In this paper, we shall provide lower and upper bounds for the functions ∑ n ≤ x z ( n ) / n , ∑ p ≤ x z ( p ) and ∑ p r ≤ x z ( p r ) .

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