On the sum of the first n values of the Euler function

2014 ◽  
Vol 163 (3) ◽  
pp. 199-201 ◽  
Author(s):  
R. Balasubramanian ◽  
Florian Luca ◽  
Dimbinaina Ralaivaosaona
Keyword(s):  
Euler Function

Carmichael Numbers with a Square Totient

2009 ◽  
Vol 52 (1) ◽  
pp. 3-8 ◽  
Author(s):  
W. D. Banks
Keyword(s):  
Euler Function ◽  
Carmichael Numbers

AbstractLet φ denote the Euler function. In this paper, we show that for all large x there are more than x0.33 Carmichael numbers n ⩽ x with the property that φ(n) is a perfect square. We also obtain similar results for higher powers.

On a sum involving the Euler function

2020 ◽  
Vol 211 ◽  
pp. 199-219 ◽  
Author(s):  
Wenguang Zhai
Keyword(s):  
Euler Function

scholarly journals Remark on Smith’s result on a divisor problem in arithmetic progressions

1985 ◽  
Vol 98 ◽  
pp. 37-42 ◽  
Author(s):  
Kohji Matsumoto
Keyword(s):  
Asymptotic Formula ◽  
Error Term ◽  
Divisor Problem ◽  
Arithmetic Progressions ◽  
Euler Function

Let dk(n) be the number of the factorizations of n into k positive numbers. It is known that the following asymptotic formula holds: where r and q are co-prime integers with 0 < r < q, Pk is a polynomial of degree k − 1, φ(q) is the Euler function, and Δk(q; r) is the error term. (See Lavrik [3]).

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 $n$.

scholarly journals The Solutions of Generalized Euler Function Equation φ_2(n-φ_2 (n))=2ω(n)

2021 ◽  
pp. 15-22
Author(s):  
Xu Yifan ◽  
Shen Zhongyan
Keyword(s):  
Positive Integer ◽  
Upper Bound ◽  
Euler Function ◽  
Prime Factors ◽  
Integer Solutions

By using the properties of Euler function, an upper bound of solutions of Euler function equation  is given, where  is a positive integer. By using the classification discussion and the upper bound we obtained, all positive integer solutions of the generalized Euler function equation  are given, where is the number of distinct prime factors of n.

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