Newbetuts

Arithmetical Functions Sum, $\sum\limits_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum\limits_{d|n}\tau(d)\phi(\frac{n}{d})$

$ (1...1)(n) =\# \{(m_1,...,m_k)\in (\mathbb{N}\setminus \{0\})^k : m_1...m_k = n\} $

Proving that $\omega(N)\neq4$ for an odd perfect number $N$ by hand

When does $\gcd(m,\sigma(m^2))$ equal $\gcd(m^2,\sigma(m^2))$? What are the exceptions?

On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number

Euler phi function: $\sum_{n=1}^{N}\sum_{d\mid n}d\cdot\phi(d)$

Polynomials whose fractional part behaves like a logarithm

Let $f:\Bbb{N}\to\Bbb{C}$ denotes the indicator function of squares. Express it in terms of Mobious function $\mu$.

Does the following lower bound improve on $I(q^k) + I(n^2) > 3 - \frac{q-2}{q(q-1)}$, where $q^k n^2$ is an odd perfect number? - Part II

Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

Help with "A Simpler Dense Proof regarding the Abundancy Index."

Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Intuitive basis of Mobius inversion?

How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$

Sums of the form $\sum_{d|n} x^d$

Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}$.

New posts in arithmetic-functions

a formula involving order of Dirichlet characters, $\mu(n)$ and $\varphi(n)$

Some regularity in the prime decomposition

Identity with nested sum taken over divisors of $\gcd$'s

Arithmetical Functions Sum, $\sum\limits_{d|n}\sigma(d)\phi(\frac{n}{d})$ and $\sum\limits_{d|n}\tau(d)\phi(\frac{n}{d})$

$ (1...1)(n) =\# \{(m_1,...,m_k)\in (\mathbb{N}\setminus \{0\})^k : m_1...m_k = n\} $

Proving that $\omega(N)\neq4$ for an odd perfect number $N$ by hand

When does $\gcd(m,\sigma(m^2))$ equal $\gcd(m^2,\sigma(m^2))$? What are the exceptions?

On the inequality $I(q^k)+I(n^2) \leq \frac{3q^{2k} + 2q^k + 1}{q^k (q^k + 1)}$ where $q^k n^2$ is an odd perfect number

Euler phi function: $\sum_{n=1}^{N}\sum_{d\mid n}d\cdot\phi(d)$

Polynomials whose fractional part behaves like a logarithm

Let $f:\Bbb{N}\to\Bbb{C}$ denotes the indicator function of squares. Express it in terms of Mobious function $\mu$.

Does the following lower bound improve on $I(q^k) + I(n^2) > 3 - \frac{q-2}{q(q-1)}$, where $q^k n^2$ is an odd perfect number? - Part II

Is there a "nice" formula for $\sum_{d|n}\mu(d)\phi(d)$?

Help with "A Simpler Dense Proof regarding the Abundancy Index."

Find all positive integers $n$ such that $\phi(n)+\sigma(n)=2n$.

For all $n$ there exists $x$ such that $\varphi(x)<\varphi(x+1)<\ldots<\varphi(x+n)$

Intuitive basis of Mobius inversion?

How to prove $ \prod_{d|n} d= n^{\frac{\tau (n)}{2}}$

Sums of the form $\sum_{d|n} x^d$

Show that $\sum\limits_{n\in \mathbb N} \frac{2^{\omega(n)}}{n^s}=\frac{\zeta^2(s)}{\zeta(2s)}$.