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For a fixed integer $N\in\mathbb{N}$ consider the multi-set $A_2(N)$ of decimal digits of $2^n$, for $n=1,2,\dots,N$. For example, $$A_2(8)=\{2,4,8,1,6,3,2,6,4,1,2,8,2,5,6\}.$$ Similarly, define the m …

Denote $\pmb{a}=(a_1,\dots,a_d)\in\mathbb{R}^d$ and consider the set $$\mathcal{E}_d=\{\pmb{a}\in\mathbb{R}^d: \text{each root $\xi$ of $x^d+a_dx^{d-1}+\cdots+a_2x+a_1=0$ lies in $\vert\xi\vert<1$}\}. …

While working with multi-zeta functions, I encountered the below (experimental) value for a certain evaluation (in a limit sense). Notice first this well-known fact in context $$\sum_{n,m\geq1}\frac1{ …

Define the sequence $$a_s=(-1)^{\binom{s-1}2}\left(\frac{\pi}2\right)^s\frac1{2\cdot s!}\begin{cases} s\,E_{s-1}, \qquad \text{if $s$ is odd} \\ 2^{2s}B_s, \qquad \,\,\text{if $s$ is even};\end{cases} …

Let $\nu_2(a)$ be the $2$-adic valuation of an integer $x$, i.e. the largest power $t$ such that $2^t$ divides $x$. Define the operator $D=x\frac{d}{dx}$ and the polynomial $\Phi_k(x)=\frac{x^{k+1}-1} …

Lagrange's four square_theorem states that every positive integer $N$ can be written as a sum of four squares of integers. At present, let's focus only on positive integer summands; that is, $N=a_1^2+ …

Recall the classical $\theta(q):=\prod_{k=1}^{\infty}(1-q^k)$ and define the sequences $a_n$ and $b_n$ by $$\frac{\theta^3(q)}{\theta(q^3)}=\sum_{n=0}^{\infty}a_nq^n \qquad \text{and} \qquad F(q):=\ …

Define the $m$-th iterated harmonic sums in the manner: $\bar{H}_0(n):=1$ and for $m\geq1$ by $$\bar{H}_m(n):=\sum_{k=1}^n\frac{\bar{H}_{m-1}(k)}k.$$ For example, $\bar{H}_1(n)=\sum_{k=1}^n\frac1k$ …

If $\sigma_k(n)=\sum_{d\vert n} d^k$, denote $$F_1(q)=\sum_{n\geq1}\sigma_1(n)\,q^n \qquad \text{and} \qquad F_3(q)=\sum_{n\geq1}n\cdot\sigma_2(n)\,q^n.$$ QUESTION. Assume the prime $p$ is either $2 …

I came across this problem while doing some simplifications. So, I like to ask QUESTION. Is there a closed formula for the evaluation of this series? $$\sum_{(a,b)=1}\frac{\cos\left(\frac{a}b\right)} …

Given a positive integer $n$, consider the diophantine equation $4x^2+y^2+4x+y=2n$ with solutions in non-negative integers $x$ and $y$. Define the proportion $$\delta_n=\frac{\#\{(x,y)\in\mathbb{Z}^2_ …

The ordinary, exponential and Dirichlet generating functions for a sequence $\{a_n\}_{n\geq0}$ are given (at least on the formal side), respectively, by $$F(x)=\sum_{n\geq0}a_nx^n, \qquad E(x)=\sum_{n …

Question. Can this number be expressed in terms of classical values? $$\sum_{n,m=1}^{\infty}\frac1{(n^2+m^2)^{\frac32}}=1.056348517615643291\dots$$ UPDATE. I'm encouraged by Noam, Kevin and Igo …

The integer partition function $p(n)$ has a generating function given by $$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$ with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem …

Let $\{x_i\}:=\{x_1=5, x_2=13, x_3=29, x_4=37, x_5=45, \dots \}$ be the sequence of those positive integers of the form $$ p^{4\alpha+1}n^2$$ in increasing order where $p\equiv 5\pmod 8$ is prime a …