User Alexander Kalmynin

23 votes

Does the average primeness of natural numbers tend to zero?

answered Apr 8, 2019 at 14:09

22 votes

Accepted

The constant $e$ represented by an infinite series

answered Jun 15, 2021 at 7:31

22 votes

Find all positive integers $n$ such that $n+\tau{(n)}=2\varphi{(n)}$

answered Mar 20, 2018 at 17:48

22 votes

Accepted

Number of 1's in binary expansion of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$

answered Jun 27, 2021 at 22:40

20 votes

Accepted

Under a condition, $\frac{1}{b } = \sum_{n=1}^{\infty}\frac{1}{a_{n}}$ will never happen

answered Jun 7, 2021 at 8:49

19 votes

Accepted

Are there any papers about this observation of the distribution of the zeros of the zeta function?

answered May 31, 2021 at 3:24

18 votes

Accepted

Singularities of power series

answered Jan 19, 2020 at 22:15

18 votes

Accepted

Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?

answered May 19, 2023 at 17:22

14 votes

Non trivial zeros of Riemann zeta function

answered Jun 16, 2021 at 19:12

14 votes

Sum of the reciprocals of radicals

answered Nov 15, 2018 at 16:59

14 votes

Probability that product is a perfect square

answered Aug 20, 2017 at 17:32

13 votes

A sum involving Euler totient function

answered Aug 13, 2017 at 21:04

13 votes

Accepted

on a strange character sum

answered May 31, 2021 at 23:17

11 votes

Are there more than two rational solutions to $a^b= b^a$?

answered May 23, 2019 at 16:13

11 votes

Accepted

Asymptotics of $\operatorname{lcm} ((2-1), (3-1), (5-1), (7-1), (11-1), \dotsc, p_n-1 )$

answered Sep 16, 2021 at 22:03

10 votes

Can $2^n\pm n$ with $n>2$ be a triangular number?

answered yesterday

10 votes

Accepted

$3\times 3$ magic squares consisting of entries of a dense set $D\subseteq \mathbb{N}$

answered Aug 2, 2022 at 8:42

10 votes

Must Mersenne numbers be divisible by arbitrary large primes with exponent one?

answered Jun 4, 2021 at 11:06

9 votes

Accepted

Covering the primes with pairs of consecutive integers

answered Apr 9, 2019 at 21:54

9 votes

Accepted

Eisenstein series $E_4(z)$ and $E_6(z)$ algebraically independent over $\mathbb{C}$

answered Oct 2, 2017 at 23:44

9 votes

Accepted

Condition for $8p+1$ divides $(2^p+1)/3$?

answered Aug 9, 2022 at 1:00

8 votes

Accepted

Are there infinitely many nonzero Euler quotients $a(n)=\frac{2^{\phi(n)}-1}{n} \bmod n$?

answered Jul 16, 2022 at 11:30

8 votes

Accepted

Prove $ \sum_{i=0}^{2a+1} {2a+1 \choose i} B_{2a+1-i} [ (n+1)^i+(-n)^i ] =0 $ for Bernoulli numbers $B_{n}$

answered Jun 4, 2021 at 17:25

8 votes

Accepted

Is it true that $\det\big[\sin 2\pi\frac{(j-k)^2}p\big]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}}$ for each prime $p\equiv3\pmod4$?

answered Jun 13, 2021 at 0:43

7 votes

Accepted

Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity?

answered Jun 15, 2021 at 8:36

7 votes

Accepted

Factorising n! fitting $x^\frac{n}{k}$

answered Aug 24, 2022 at 9:16

7 votes

Accepted

A conjecture concerning the equation $\sigma\left(\square\right)=\text{prime}$

answered Jun 11, 2022 at 20:18

7 votes

Accepted

Fourier transform of periodic distributions

answered Jul 14, 2021 at 11:26

7 votes

Accepted

Given the integral. What's the relation between $I_{n+1}(t)$ and $I_n(t)$?

answered Jul 15, 2021 at 18:47

6 votes

Accepted

Upper bound for maximal gap between consecutive numbers consisting only $4k+1$ primes

answered Aug 18, 2021 at 15:14

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64 Answers

Does the average primeness of natural numbers tend to zero?

The constant $e$ represented by an infinite series

Find all positive integers $n$ such that $n+\tau{(n)}=2\varphi{(n)}$

Number of 1's in binary expansion of $a_n = \frac{2^{\varphi(3^n)}-1}{3^n}$

Under a condition, $\frac{1}{b } = \sum_{n=1}^{\infty}\frac{1}{a_{n}}$ will never happen

Are there any papers about this observation of the distribution of the zeros of the zeta function?

Singularities of power series

Can the equation $1+z+z^2=z^n$ for natural $n$ have multiple complex roots $z$?

Non trivial zeros of Riemann zeta function

Sum of the reciprocals of radicals

Probability that product is a perfect square

A sum involving Euler totient function

on a strange character sum

Are there more than two rational solutions to $a^b= b^a$?

Asymptotics of $\operatorname{lcm} ((2-1), (3-1), (5-1), (7-1), (11-1), \dotsc, p_n-1 )$

Can $2^n\pm n$ with $n>2$ be a triangular number?

$3\times 3$ magic squares consisting of entries of a dense set $D\subseteq \mathbb{N}$

Must Mersenne numbers be divisible by arbitrary large primes with exponent one?

Covering the primes with pairs of consecutive integers

Eisenstein series $E_4(z)$ and $E_6(z)$ algebraically independent over $\mathbb{C}$

Condition for $8p+1$ divides $(2^p+1)/3$?

Are there infinitely many nonzero Euler quotients $a(n)=\frac{2^{\phi(n)}-1}{n} \bmod n$?

Prove $ \sum_{i=0}^{2a+1} {2a+1 \choose i} B_{2a+1-i} [ (n+1)^i+(-n)^i ] =0 $ for Bernoulli numbers $B_{n}$

Is it true that $\det\big[\sin 2\pi\frac{(j-k)^2}p\big]_{1\le j,k\le p-1}=-\frac{p^{(p-1)/2}}{2^{p-1}}$ for each prime $p\equiv3\pmod4$?

Is this Goldbach conjecture related quantity equal to the number of Goldbach decompositions up to a bounded quantity?

Factorising n! fitting $x^\frac{n}{k}$

A conjecture concerning the equation $\sigma\left(\square\right)=\text{prime}$

Fourier transform of periodic distributions

Given the integral. What's the relation between $I_{n+1}(t)$ and $I_n(t)$?

Upper bound for maximal gap between consecutive numbers consisting only $4k+1$ primes