0
votes
5answers
118 views

For every positive integer n greater than $2$, $\phi(n)$ is an even integer.

Theorem: For every positive integer n greater than $2$, then $\phi(n)$ is an even integer. I know this theorem and the same is used much, but I was curious how it would be to demonstrate it, show ...
1
vote
2answers
88 views

If $(m,n)=1$ then $m^{\phi(n)}+n^{\phi(m)}\equiv0\pmod {mn}$

Show that if $(m,n)=1$ then $m^{\phi(n)}+n^{\phi(m)}\equiv0\pmod {mn}$ I tried, $(m,n)=1$ then, Euler theorem we have $m^{\phi (n)}\equiv1\pmod n$ and $n^{\phi (m)}\equiv1\pmod m$. But I could ...
-1
votes
5answers
112 views

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$

Solve for $m\in\mathbb{N}$ the equation $\phi (m)=12$ I found (by trial) $m=\{13,21,26,28,36\}$, but do not know if misinterpreted the problem, but actually I suppose I have to find an equation ...
3
votes
3answers
75 views

Prove that for all $n\in\mathbb{N}$, $\frac{s(n)}{d(n)}\geq \sqrt n$

Prove that for all $n\in\mathbb{N}$ $$\frac{s(n)}{d(n)}\geq \sqrt n$$ where $s(n) = \sum_{d|n} d$ and $d(n) = \sum_{d|n} 1$. Being honest, study some time arithmetic functions, and can not ...
0
votes
1answer
32 views

Demonstration in arithmetic function

I need help to know (in detail) how to prove that the product of two multiplicative arithmetic functions is a multiplicative arithmetic function. $$$$$f(n)$ and $g(n)$ are functions multiplicative, ...
3
votes
1answer
54 views

Is there a friendly pair $\{a, b\}$ where both $a$ and $b$ are odd?

Let $\sigma(x)$ denote the sum of the divisors of a positive integer $x$. $\{y, z\}$ is said to be a friendly pair if $$I(y) = I(z),$$ where $I(x) = \sigma(x)/x$ is the abundancy index of $x$. As ...
0
votes
0answers
56 views

Does there exist a counterexample to the inequality $I(n) < 2{\left(\frac{n}{n + 1}\right)}^2$, if $n$ is odd and deficient?

If $n$ is odd and deficient, does there exist a counterexample to the inequality $$I(n) < 2{\left(\frac{n}{n + 1}\right)}^2,$$ where $$I(x) = \frac{\sigma(x)}{x}$$ is the abundancy index of ...
0
votes
0answers
42 views

What proportion of the positive integers satisfy $I(n^2) < (1 + \frac{1}{n})I(n)$, where $I(x)$ is the abundancy index of $x$?

Let $\sigma(x)$ denote the classical sum-of-divisors function, and let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. My question is this: What proportion of ...
0
votes
1answer
83 views

What proportion of the positive integers satisfy $I(n) < \frac{2n}{n + 1} \leq I(n^2)$ < 2?

Let $$I(x) = \frac{\sigma(x)}{x}$$ be the abundancy index of the positive integer $x$. Note that $\sigma(x)$ is the classical sum-of-divisors function. For example, $$\sigma(12) = 1 + 2 + 3 + 4 + ...
1
vote
1answer
69 views

What proportion of the natural numbers satisfy the following inequalities?

Let $\sigma_1(n)$ be the sum of the divisors of $n \in \mathbb{N}$, and let $$I(n) = \frac{\sigma_1(n)}{n}$$ be the abundancy index of $n$. What proportion of the natural numbers satisfy the ...
0
votes
0answers
96 views

Can an odd perfect number be divisible by 101?

Preamble - This question is an offshoot from the following earlier questions here at MSE: Can an odd perfect number be divisible by 825? Can an odd perfect number be divisible by 165? Odd perfect ...
1
vote
3answers
122 views

Are there any integer solutions to $\gcd(\sigma(n), \sigma(n^2)) = 1$ other than for prime $n$?

A good day to everyone! Are there any integer solutions to $\gcd(\sigma(n), \sigma(n^2)) = 1$ other than for prime $n$ (where $\sigma = \sigma_1$ is the sum-of-divisors function)? Note that, if $n = ...
2
votes
1answer
72 views

how to prove $f$ is an arithmetic function with this property $\sum_{d\mid n} f(d)=n^2$

how to prove $f$ is an arithmetic function with this property $$\sum_{d\mid n} f(d)=n^2$$ Arithmetic function
4
votes
2answers
86 views

Prove: $\sum_{k<n, (k,n)=1} k= \frac{1}{2}n \varphi (n)$

Prove: $\sum_{k<n, (k,n)=1}k = \frac{1}{2}n \varphi (n)$ I have had strep throat and missed the lecture discussing properties of the Euler function. Any help in solving this is appreciated. ...
1
vote
2answers
74 views

Is Wiki wrong on Dirichlet Chararcters Modulo $10$?

Wiki says: Modulus 10 There are $\phi(10)=4$ characters modulo $10$. Note that $χ$ is wholly determined by $\chi(3)$, since $3$ generates the group of units modulo $10$. I can ...
1
vote
1answer
160 views

Function that counts the number of divisors of a natural number?

Let Function $f(n)$ be formally defined for natural numbers such that it gives number of distinct divisors of the number n (n and 1 included) For example, $f (12)=6$, then what is a quick way to ...
1
vote
0answers
111 views

Generalizing a result on sums involving Euler's function

Motivation: It's known that there is a constant $0<K$ such that for any natural number $N$, $KN\leq \frac{\varphi(1)}{1}+\frac{\varphi(2)}{2}+\cdots+\frac{\varphi(N)}{N}$ (with $\varphi$ being ...
4
votes
1answer
408 views

On sums involving Euler's totient function

I've been struggling with the following claim without being able to prove it, so your help would be highly appreciated: Let $\varphi(n)$ be Euler's totient function. Show that there is a constant ...