**3**

votes

**1**answer

206 views

### Estimation of $\sum_{n \leq x} \frac{k(n)}{n}$ , with $k(n)$ the squarefree kernel

I came across a poblem where they ask you to find an estimation of $\sum_{n \leq x} \frac{k(n)}{n}$, with $k(n) = \prod_{p \mid n} p$ the squarefree kernel of $n$, with an error term of $O(\sqrt{x})$.
...

**0**

votes

**0**answers

120 views

### On the number N(x,y) of those integers n<x, with squarefree core k(n)<y

I'm asking something that may be trivial for those who are deeply into Analytic Number Theory, but unfortunately I'm still not into that set.
The core $k(n)$ of an integer $n$ is the product of all ...

**3**

votes

**2**answers

433 views

### A set with not too many integers of the form $\alpha \beta^n + r$

Consider the following (easy) lemma.
Lemma. There is a subset $Q$ of the positive integers and a fixed constant $N > 0$ such that
1)$Q$ has positive asymptotic density and
2)for each ...

**28**

votes

**3**answers

3k views

### $\prod_{n=1}^{\infty} n^{\mu(n)}=\frac{1}{4 \pi ^2}$

When I tested this in Mathematica, I had expected it to say it did not converge. However, I got this:
$$\prod_{n=1}^\infty n^{\mu(n)}=\frac{1}{4 \pi ^2}$$
Note: this is the reciprocal of (3) ...

**14**

votes

**4**answers

2k views

### are there infinitely many triples of consecutive square-free integers?

The title says it all ... Obviously, any such triple must be of the form
$(4a+1,4a+2,4a+3)$ where $a$ is an integer. Has this problem
already been studied before ? The result would follow from ...

**5**

votes

**1**answer

407 views

### Integer polynomial (of degree >1) all of whose values are square-free

Is there an integer polynomial $ A \in {\mathbb Z} [ X ]$ of degree $d\geq 2$ such that for any integer $n\in {\mathbb Z}$ , $ A(n) $ is a square-free integer?