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15 votes
4 answers
2k views

Square roots and prime numbers

Definitions: Here I present a novel conjecture using basic mathematical tools like the sum of the divisors of an integer $n$ called $\sigma(n)$, the sum of the squares of the positive divisors of n ...
Sulfura's user avatar
  • 127
3 votes
2 answers
581 views

Approximation of partial sum over prime omega function

I asked the question in Math StackExchange. Link: https://math.stackexchange.com/questions/4765476/approximation-of-partial-sum-over-prime-omega-function I haven't got any response yet. Here are the ...
piepie's user avatar
  • 221
8 votes
1 answer
205 views

Are there infinite numbers of the form $\sigma_1(n)=\sigma_1(m)=p$, or is there only one?

I put forward a hypothesis in number theory, it is as follows.$ \sigma_1(n)=\sigma_1(m)=p$, where $\sigma_1$ is the divisor sum function, $n,m\in \mathbb N$, and $p$ is prime. I recently noticed and ...
Arsen Vardanyan's user avatar
6 votes
1 answer
392 views

Arithmetic properties of positively reduced $2\times 2$-matrices

Call a $2\times 2$ matrix with coefficients in $\{0,1,2,3,\ldots\}$ positively reduced if any row or column reduction (given by replacing a row/column by itself minus the other row/column) produces at ...
Roland Bacher's user avatar
1 vote
1 answer
347 views

On equations with arithmetic functions [closed]

Is this good topic for research: equations with arithmetic functions, for example equations like $\varphi(n)=\sigma(n)$ or $\varphi(n)+\sigma(n)=d(n)$ ? If Anyone here have an advise please tell me ...
Omega's user avatar
  • 31
-10 votes
1 answer
555 views

Arithmetic billiards, prime numbers and the Goldbach conjecture

I've edited the following post on Mathematics Stack Exchange, (now closed, at that date I'm suspended) with identifier 4510963, please let me to know if you've some doubt or I can improve the post. On ...
user142929's user avatar
1 vote
1 answer
123 views

Periodic sequences of integers generated by $a_{n+1}=\frac{\operatorname{rad}(pa_{n})}{p}+\frac{\operatorname{rad}(qa_{n-1})}{q}$

Let's define the radical of the positive integer $n$ as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$ and consider the sequence $$a_{n+1}=\frac{\operatorname{rad}(p\cdot a_{n})...
Augusto Santi's user avatar
0 votes
0 answers
68 views

Around similar inequalities than an inequality due to Nicolas, that involve products of consecutive Ramanujan primes

This is cross-posted (and this post is a version to ask just around the veracity of Conjecture 1) as the post with identifier 3594907 and same title), that I've edited on Mathematics Stack Exchange ...
user142929's user avatar
1 vote
0 answers
153 views

A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
user142929's user avatar
3 votes
1 answer
134 views

Is it possible to find an estimate of $\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$?

Is it possible to find an estimate of the summation $$s(n)=\sum_{k=1}^n\frac1{\varphi(k\cdot p_k)}$$ where $\varphi(n)$ is the totient function and $p_k$ the k-th prime? The corresponding series seems ...
Augusto Santi's user avatar
0 votes
0 answers
89 views

A similar inequality for the Dedekind psi function, than an inequality stated by Schinzel

I would like to ask about the next question that seems to me interesting. I know an article that was written by Andrzej Schinzel in which he stated Lemma 2. In this post we denote the Dedekind psi ...
user142929's user avatar
7 votes
1 answer
231 views

The asymptotic of $|\{1\leq n\leq x|\gcd(n,S(n))=1\}|$, with $S(n)$ the sum of remainders, and get idea for other miscellany problem

Let $n\geq 1$ be an integer. In this post we denote the sum of remainders function as $$S(n)=\sum_{k=1}^n n \bmod k,$$ for example $S(1)=S(2)=0+0$ and $S(5)=0+1+2+1+0=4$. In the literature there are ...
user142929's user avatar
0 votes
0 answers
143 views

Given $\,m=\prod_k {p_k}^{\alpha_k}\,$ and the function $\,g(m)=\sum_k \alpha_k(p_k-1)^2$, find all solutions of the equation $\,g(2n)=n$

Let's consider the unique decomposition of a natural number $\,m\,$ into its prime factors: $$\prod_k {p_k}^{\alpha_k}$$ Then, let's define the following arithmetic function (completely additive) $\,g:...
Augusto Santi's user avatar
4 votes
1 answer
530 views

Reference for inequality for $\sum\limits_{d \mid n}\frac{\log d}{d}.$

Let $f(n)=\sum\limits_{d \mid n}\frac{\log d}{d}.$ It is not hard to see that $f(n)\ll(\log\log n)^2$. Is there any reference for this inequality? EDT 1: A possible answer is Analysis of the ...
Alexey Ustinov's user avatar
8 votes
1 answer
427 views

Goldbach's conjecture for the Liouville function

Is it true that for every even integer $N > 2$, there exist positive integers $a,b$ such that $a + b = N$ and $\lambda(a) = \lambda(b) = -1$ ? Here $\lambda$ is the Liouville function.
Pablo's user avatar
  • 11.3k
0 votes
0 answers
90 views

Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^{\frac{{p_1}-3}{p_2}}-1)\equiv 0 \bmod p_1$ with $p_1,p_2\equiv 3\bmod 4$?

let $p_1$ and $p_2$ be positive primes such that $p_1,p_2 \equiv 3\bmod 4$ and $\phi$ is the Euler totiont function , I want to find the Set of primes $p_{1}\equiv 3 \bmod p_{2}$ such that $\phi(2^...
zeraoulia rafik's user avatar
5 votes
2 answers
314 views

Congruences for the non-divisors of Euler's $\phi(n)$

If $n$ is composite, then $\phi(n) < n-1$: hence, there is at least one number $d$ which does not divide $\phi(n)$ but divides$(n-1)$. We shall call $d$ the totient divisor of $n$. The purist will ...
Nilotpal Kanti Sinha's user avatar
5 votes
0 answers
229 views

Can we write each positive integer as $x^2+y^2+\varphi(z^2)$?

As odd squares are congruent to $1$ modulo $8$, any integer of the form $4^k(8m+7)$ with $k,m\in\mathbb N=\{0,1,2,\ldots\}$ cannot be written as the sum of three squares. To avoid such congruence ...
Zhi-Wei Sun's user avatar
  • 15.6k
4 votes
1 answer
322 views

A special kind of multiplicative function $f: \mathbb N \to \mathbb N$ such that $f(p)=p+k$ for all odd prime $p$, where $k>1$ is a fixed odd integer

For which odd positive integer $k$, can we find a multiplicative function $f: \mathbb N \to \mathbb N$ satisfying the following conditions : $f(p)=p+k$ for all large enough odd prime $p$ and the set $...
user avatar
1 vote
0 answers
233 views

Primes approximated by eigenvalues?

Let the matrix $T$ be defined by: $$\displaystyle T(n,k) = -\varphi^{-1}(\operatorname{GCD}(n,k))$$ where $\varphi^{-1}$ is the Dirichlet inverse of the Euler totient function. $$\varphi^{-1}(n) = \...
Mats Granvik's user avatar
  • 1,183
3 votes
0 answers
182 views

Mersenne number with small Carmichael function

Let $\lambda(\cdot)$ be the Carmichael function. I'm trying to understand the magnitude of the smallest values of $\lambda(2^n - 1)$, when $n$ runs over the positive integers. Precisely my question is:...
user avatar
3 votes
0 answers
443 views

Infinite sums with Mobius Inversion : can we have uniform convergence of inversion formula?

My question is on Mobius inversion formula convergence/properties when used with infinite sums of function. Lets consider (on $\mathbb{R}^{+}$): $$S(x)= \sum\limits_{n=1}^{\infty} f(nx)$$ We call $...
Bertrand's user avatar
  • 1,199
2 votes
1 answer
1k views

A formula combining Euler $\phi$ and $\gcd$

Let us fix a natural number $N>1$ and $a_1, \ldots, a_n$ natural numbers satisfying $0 \leq a_i < N$, with the property that $1+ \sum a_i$ is divisible by $N$. Let $\phi$ be the Euler totient ...
calc's user avatar
  • 283
7 votes
0 answers
332 views

$n\varphi(n)\equiv 2\pmod{\sigma(n)}$ as a primality test

It is known from Subbarao, "On two congruences for primality" that $n>22$ is a prime iff $$n\sigma(n)\equiv 2\pmod{\varphi(n)},$$ where $\varphi(n)$ is Euler's function and $\sigma(n)$ is sum of ...
Alexey Ustinov's user avatar