All Questions
16
questions
4
votes
2
answers
166
views
Asymptotics of $p_k$-adic valuation of the sum of the divisors of the $n$-th primorial
Given this product:
$$a(n) = \prod_{k=1}^{n} (1+p_k)$$
where $p_k$ is the $k$-th prime number and which can be interpreted also as the sum of the divisors of the $n$-th primorial (OEIS A054640), is ...
- 2,103
3
votes
1
answer
119
views
Length of this representation increases really slowly?
$$\def\'{\text{'}}\def\len{\operatorname{len}}$$
A recent Code Golf challenge introduced a "base neutral numbering system". Here I present a slightly modified version, but the idea is the ...
- 562
18
votes
2
answers
630
views
$\lim_{n \to \infty}\frac{1}{n}\sum_{k = 1}^nf(k)$ where $f(n)$ is the largest prime factor exponent?
Let $f(n)$ the be largest exponent among exponents of the prime factor of $n$. E.g. $f(80) = 4$ since $80 = 2^4.5$ and the prime factor of $80$ which has the largest exponent is $2$ which occurs with ...
- 20.8k
6
votes
0
answers
383
views
What is the most frequent largest prime factor of the numbers between two primes?
Let $p_n$ be the $n$-th prime and $l_n, n \ge 2$ be the largest of all the prime factors of the composite numbers between $p_n$ and $p_{n+1}$. Since there are infinitely many prime gaps, each of these ...
- 20.8k
1
vote
2
answers
227
views
Equations involving particular values of the Dedekind psi function and powers of the kernel function
In this post we denote the Dedekind psi function as $\psi(m)$ for integers $m\geq 1$. As reference I add the Wikipedia Dedekind psi function, and [1]. One has the definition $\psi(1)=1$, and that the ...
1
vote
0
answers
91
views
Generalization on a feature of 21
Let $n,m \in \mathbb{N}$
$$n=\prod_{i=1}^{r}p_{i}^{a_i}$$
where $p_i$ are prime factors and
$f$ , $g$ and $h$ are the functions
$$f(n,m)=\sum_{j=1}^{n}j^m$$
And
$$g(n)=\sum_{i=1}^{r}a_i.p_i$$
If we ...
- 2,697
7
votes
1
answer
298
views
Are there infinitely many zeroes of $\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) $?
Let $\mu(n)$ be the Möbius function and $S(x)$ be the number of positive integers $n \le x$ such that
$$
\sum_{r = 1}^{n-1} \mu(r)\gcd(n,r) = 0
$$
My experimental data for $n \le 6 \times 10^5 $...
- 20.8k
6
votes
0
answers
149
views
Density of Numbers with Exactly One Prime Factor of Multiplicity 1
Let $S$ be the set of positive integers $n$ with the property that exactly one prime factor of $n$ has multiplicity $1$ and every other prime factor has multiplicity greater than $1$ (to be clear, $S$ ...
- 2,726
0
votes
1
answer
65
views
Understanding part of derivation of Chebychev's Theorem
I cannot understand this result from pages 17–18 of Tenenbaum and Mendes's The Prime Numbers and Their Distribution on how the summation of $\frac{x\log(2)}{2^j}+O(\log(x))$ results in $2x \log(2) + O(...
- 1,379
2
votes
1
answer
101
views
Are there infinitely many primes of the form $2\,m^{\operatorname{gpf}(m)}+1$ when $m$ runs over positive integers?
Let $n\geq 1$ an integer, in this post we denote the greatest prime dividing $n$ as $\operatorname{gpf}(n)$.
See it you want the article from MathWorld Greatest Prime Factor.
While I was writing ...
user243301
3
votes
1
answer
77
views
On the equation $\sigma(m)=105k$ over odd integers $m\geq 1$, and the deficiency of its solutions
In this post we denote the sum of positive divisors of a natural $n\geq 1$ as $$\sigma(n)=\sum_{d\mid n}d.$$
We consider integers $m\geq 1$ and $k\geq 1$ where $m\equiv 1\text{ mod }2$ (thus $m$ is ...
user243301
1
vote
1
answer
243
views
Square-free integers in the sequence $n^{\operatorname{rad}(n)}+\operatorname{rad}(n)+1$
For integers $n\geq 1$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an ...
user243301
4
votes
2
answers
308
views
On questions involving the radical of an integer and different number theoretic functions: the Euler's totient function
We denote for an integer $n>1$ its square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$
with the definition $\operatorname{rad}(1)=1$. You can see this ...
user243301
5
votes
1
answer
224
views
An approximation for $1\leq n\leq N$ of the number of solutions of $2^{\pi(n)}\equiv 1\text{ mod }n$, where $\pi(x)$ is the prime-counting function
We denote the prime-counting function with $\pi(x)$ and we consider integer solutions $n\geq 1$ of the congruence $$2^{\pi(n)}\equiv 1\text{ mod }n.\tag{1}$$
Then the sequence of solutions starts as $$...
user243301
2
votes
1
answer
105
views
Square-free integers in the sequence $\lambda+\prod_{k=1}^n(\varphi(k)+1)$, where $\lambda\neq 0$ is integer
While I was exploring the squares in the sequence defined for integers $n\geq 1$
$$\prod_{k=1}^n(\varphi(k)+1),\tag{1}$$
where $\varphi(m)$ denotes the Euler's totient function I wondered a different ...
user243301