The Average of the Functions Associated Euler Function in Long Intervals

2021 ◽  
Vol 11 (05) ◽  
pp. 832-840
Author(s):  
美玲 张
Keyword(s):  
Euler Function

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 $n$.

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