scholarly journals WHEN THE SUM OF ALIQUOTS DIVIDES THE TOTIENT

2007 ◽  
Vol 50 (3) ◽  
pp. 563-569
Author(s):  
William D. Banks ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractLet $\varphi(\cdot)$ be the Euler function and let $\sigma(\cdot)$ be the sum-of-divisors function. In this note, we bound the number of positive integers $n\le x$ with the property that $s(n)=\sigma(n)-n$ divides $\varphi(n)$.

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

Amicable pairs with few distinct prime factors

2016 ◽  
Vol 12 (07) ◽  
pp. 1725-1732
Author(s):  
Florian Luca ◽  
M. Tip Phaovibul
Keyword(s):  
Prime Factors ◽  
Positive Integers ◽  
Sum Of Divisors

An amicable pair [Formula: see text] is a pair of distinct positive integers [Formula: see text] such that [Formula: see text], where [Formula: see text] is the sum of divisors function. In this note, we prove that if [Formula: see text] are amicable and [Formula: see text] is odd, then [Formula: see text], where [Formula: see text] is the number of distinct prime factors.

scholarly journals Some arithmetic functions of factorials in Lucas sequences

2021 ◽  
Vol 56 (1) ◽  
pp. 17-28
Author(s):  
Eric F. Bravo ◽  
◽  
Jhon J. Bravo ◽  
Keyword(s):  
General Class ◽  
Catalan Numbers ◽  
Arithmetic Functions ◽  
Integer Sequences ◽  
Lucas Sequences ◽  
Lucas Sequence ◽  
Positive Integers ◽  
Sum Of Divisors

We prove that if {un}n≥ 0 is a nondegenerate Lucas sequence, then there are only finitely many effectively computable positive integers n such that |un|=f(m!), where f is either the sum-of-divisors function, or the sum-of-proper-divisors function, or the Euler phi function. We also give a theorem that holds for a more general class of integer sequences and illustrate our results through a few specific examples. This paper is motivated by a previous work of Iannucci and Luca who addressed the above problem with Catalan numbers and the sum-of-proper-divisors function.

scholarly journals ON THE INDEX OF COMPOSITION OF THE EULER FUNCTION AND OF THE SUM OF DIVISORS FUNCTION

2009 ◽  
Vol 86 (2) ◽  
pp. 155-167 ◽  
Author(s):  
JEAN-MARIE DE KONINCK ◽  
FLORIAN LUCA
Keyword(s):  
Mean Value ◽  
Arbitrary Integer ◽  
Normal Order ◽  
Euler Function ◽  
Sum Of Divisors

AbstractGiven an integer n≥2, let λ(n):=(log n)/(log γ(n)), where γ(n)=∏ p∣np, denote the index of composition of n, with λ(1)=1. Letting ϕ and σ stand for the Euler function and the sum of divisors function, we show that both λ(ϕ(n)) and λ(σ(n)) have normal order 1 and mean value 1. Given an arbitrary integer k≥2, we then study the size of min {λ(ϕ(n)),λ(ϕ(n+1)),…,λ(ϕ(n+k−1))} and of min {λ(σ(n)),λ(σ(n+1)),…,λ(σ(n+k−1))} as n becomes large.

scholarly journals Power values of the product of the Euler function and the sum of divisors function

2015 ◽  
Vol 8 (5) ◽  
pp. 745-748
Author(s):  
Luis Elesban Santos Cruz ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Sum Of Divisors

scholarly journals Positive Integers Whose Euler Function Is a Power of Their Kernel Function

2006 ◽  
Vol 36 (1) ◽  
pp. 81-96 ◽  
Author(s):  
J.-M. De Koninck ◽  
F. Luca ◽  
A. Sankaranarayanan
Keyword(s):  
Kernel Function ◽  
Euler Function ◽  
Positive Integers

scholarly journals On the composition of the Euler function and the sum of divisors function

2007 ◽  
Vol 108 (1) ◽  
pp. 31-51
Author(s):  
Jean-Marie De Koninck ◽  
Florian Luca
Keyword(s):  
Euler Function ◽  
Sum Of Divisors

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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