# Tagged Questions

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### Find $\sum\frac{a(n)}{n(n+1)}$, where $a(n)$ — number of 1's in binary expansion of n. [duplicate]

Let $a(n)$ is a number of 1's in binary expansion of n, find the sum $$\sum\limits_{n=1}^{\infty}\frac{a(n)}{n(n+1)}.$$
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### How to sum this infinite series

How to sum this series: $$\frac{1}{1}+\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$$ My attempt: Multiply and divide the series by $9$ ...
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### Help with $\sum_{d\mid n}τ(d)^2=\sum_{d \mid n}τ(d)^3$

I am doing some exercises on number theory on multiplicative number theoretic functions and I have some problems with the multiplication on sums like the sum $\sum_{d\mid n}(τ(d))^2$ where $d$ is a ...
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### The sum $1!+2!+3!+…+2007!$ is not a perfect square

Today my teacher told me to prove this:- Prove that $1!+2!+3!+...+2007!=\sum_{n=1}^{2007}(n!)$ is neither a perfect square nor a perfect cube. Not getting any idea. Please help. Thanks in advance.
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### Number of palindromic numbers less than a power of $10$

I noticed that every $10^{n}$ there is a certain number of palindromic numbers that I collected in this sequence: $$S=\{a_n,a_{n+1},a_{n+2}...\}=\{10,9,90,90,900,900...\}$$ where every number $a_n$ is ...
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### How can I prove that $\frac{\sigma(n)}{n} = \sum_{(d|n)} \frac{1}{d}$ for every $n \in \mathbb{Z^{+}}$?

I want to show that $\displaystyle \frac{\sigma(n)}{n} = \sum_{(d|n)} \frac{1}{d}$ for every $n \in \mathbb{Z^{+}}$. This is essentially a basic number theory question. I am able to get to the ...
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### A different Harmonic series.

Let's call the following numbers than can be produced by playing with plus and minus: $$H_n'=\pm\frac{1}{1}\pm\frac{1}{2}\pm\frac{1}{3}\pm\cdots\pm\frac{1}{n}$$ "Harmonic kids" of $H_n$. We have a ...
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### Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
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### Finding the summation of the floor of the series identity

I would appreciate if somebody could help me with the following problem: Q: How to proof ? The number of positive divisors of $n$ is denoted by $d(n)$ ...
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### Summation involving totient function: $\sum_{d\mid n} \varphi(d)=n$ [duplicate]

Prove that:$$\sum_{d\mid n} \varphi(d)=n$$ Where $\varphi(n)$ denotes the number of positive integers $m$ less than or equal to $n$ such that $\gcd(m,n)=1$ I am lost here, any help would be ...
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### On $\lfloor\sqrt n \rfloor+ \sum_{j=1}^n \lfloor n/j\rfloor$

How do we prove that $\Big[\sqrt n \Big]+ \sum_{j=1}^n \bigg[ \dfrac nj\bigg]$ is an even integer for all $n \in \mathbb N$ ? (where $\Big[ \space \Big]$ denotes the "greatest integer" function)
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### Totient function sum over divisors

I would like to know if there is a closed form solution for $$G(n)=\sum\limits_{d\mid n}(-1)^{\frac{n}{d}}\phi(d)$$ It seems quite likely there is since $$\sum\limits_{d\mid n}\phi(d)=n$$ But I ...
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### Proving $\sum_{k=1}^n{2k-1\choose k}{2n-2k+1\choose n-k+1}=4^n-{2n+1\choose n+1}$

Some background. I was asked to find an arithmetic function $f$ such that $f*f=\mathbf 1$ where $\mathbf 1$ is the constant function 1 and $*$ denotes Dirichlet convolution. I was able to prove that ...
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### Hockey Classics at Matheletics '13

I'm trying to solve a challenge from Matheletics '13: Micheal Nobbs is organizing a training camp for identifying new talents in Indian Hockey. The camp witnessed a total of ($3K+1$) players. Each of ...
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### Is $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ for all integers $j$ and $k$, where $0\leq j\leq k$?

For all integers $j$ and $k$, where $0\leq j\leq k$, is the inequality $\frac{k!!}{j!!(k-j)!!}\leq\frac{k!}{j!(k-j)!}$ true? I have a feeling that it is and it would be helpful to me if it is, ...
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### Number of factors of summation

Let $a(n)$ be the number of $1$'s in the binary expansion of $n$. If $n$ is a positive integer, show that $$\Bigg|\sum_{k=0}^{2^n-1}(-1)^{a(k)}\times 2^k\Bigg|$$ has at least $n!$ divisors. I think ...
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### Finding a closed form for a sum involving floor function $\sum\limits_{k=1}^nk\lfloor km/n\rfloor$

Given integers $m,n$, is there a known closed form for the sum $\sum\limits_{k=1}^nk\lfloor km/n\rfloor$? and if not, is it possible to show there is no closed form for this sum? I've attempted to ...
### Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$
Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.