All Questions
Tagged with elementary-number-theory summation
482
questions
1
vote
1
answer
62
views
Max value of $p \ge 0$, $p \in \mathbb{Z}$ which makes $\frac{a_n}{3^p} \in \mathbb{N}$, when $\{a_n\}$ is an arithmetic sequence?
For $d \in \mathbb{N}$, $\{a_n\}$ is a arithmetic progression with $a_1 = 9$, and its common difference $d$.
$b_n$ is a max value of $p$ $(p \ge 0$, $p \in \mathbb{Z})$ which makes $\frac{a_n}{3^p} \...
- 343
1
vote
0
answers
59
views
Changing index of a double summation [duplicate]
How does one prove the following result, where $x$ is a three-parameter function defined on $\mathbb Z^3$? $$ \sum_{\ell=1}^{P}\sum^{\ell-1}_{i=0} x(\ell,i,\ell-i) \quad = \quad \sum^{P}_{j=1}\sum^{P-...
- 1,139
0
votes
1
answer
194
views
A sum related to the Mobius function
It is well-known that
$$\frac{\phi(n)}{n}=\sum_{d\mid n}\frac{\mu (d)}{d} \quad n\in \mathbb Z^+$$
Where $\phi $ is Euler's totient function and $\mu$ is the Mobius function. But using the formula for ...
- 4,196
1
vote
1
answer
73
views
Prove or disprove that $\sum_{k=1}^p G(\lambda^k) = ps(p)$
Prove or disprove the following: if $\lambda$ is a pth root of unity not equal to one, $G(x) = (1+x)(1+x^2)\cdots (1+x^p),$ and $s(p)$ is the sum of the coefficients of $x^n$ for $n$ divisible by $p$ ...
- 1,225
4
votes
4
answers
271
views
On swapping the order of a summation
Consider the following sum $$S=\sum_{k=1}^nd(k)$$
where $d(k)$ is the number of divisors of $k$. We can rewrite the sum like this $$S=\sum_{k=1}^n\sum_{d\mid k}^k1$$
but now how can I change the order ...
- 4,196
0
votes
1
answer
46
views
Prove that $\sum_{i = 1}^{N} 1+ (2i \bmod N) = N(N + 1) / 2$ for odd N.
I was able check by hand that for odd $N$ the $1+ (2i \bmod N)$ produces all values between $1$ and $N$ and for even $N$ there are repeats.
But I've no ideas on how to write a mathematical proof for ...
0
votes
2
answers
54
views
Compute $p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$
I need to compute $$p_{n+1} = p^2 \frac{r_1^n-r_2^n}{r_1-r_2}, n \geq 0$$ where $p=\frac{1}{2}$, $r_1= \frac{1+\sqrt5}{4}= \frac{1}{2}\varphi$ and $r_2= \frac{1-\sqrt5}{4}$.
Using properties of ...
1
vote
2
answers
83
views
Find all the integers which are of form $\dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}, a,b,c\in \mathbb{N}$, any two of $a,b,c$ are relatively prime.
I have a question which askes to find all the integers which can be
expressed as
$\displaystyle \tag*{} \dfrac{b+c}{a}+\dfrac{c+a}{b}+\dfrac{a+b}{c}$
where $a,b,c\in \mathbb{N} $ and any two of $a,b,...
- 921
4
votes
3
answers
666
views
Representing the cube of any natural number as a sum of odd numbers
I'm expanding my notes on exercises from Donald Knuth's The Art of Computer Programming, and found something rarely mentioned in the Internet, but still useful to prove Nicomachus' Theorem about the ...
- 848
2
votes
0
answers
60
views
Closed form for the sum of 2 raised to the proper divisors of a positive integer
Is there a known closed form for the sum of 2 raised to the proper divisors of a positive integer?
The problem arises in counting the number of binary sequences that can be "cycles" that ...
- 21
1
vote
0
answers
69
views
Minimum value made of the reciprocals of the first $n$ primes
Let $n$ be a positive integer , $p_k$ the k-th prime number and $a_j=-1$ or $a_j=1$ for $j=1,\cdots ,n$
What is the minimum value of $$S:=|\sum_{j=1}^n \frac{a_j}{p_j}|$$ ?
Motivation : If we ...
- 85.1k
2
votes
2
answers
153
views
$\sum_{k=0}^\infty[\frac{n+2^k}{2^{k+1}}] = ?$ (IMO 1968)
For every $ n \in \mathbb{N} $ evaluate the sum $ \displaystyle \sum_{k=0}^\infty \left[ \dfrac{n+2^k}{2^{k+1}} \right]$ ($[x]$ denotes the greatest integer not exceeding $x$)
This was IMO 1968, 6th ...
- 336
3
votes
2
answers
81
views
When is the sum of the products of any three distinct numbers less than $n$, divisible by $n$?
Define $A$ as the set of all integer triples $(a,b,c)$, such that $0<a<b<c<n$.
$$A=\left\{(a,b,c)\in \mathbb{Z^3}:1\le a<b<c\le n-1 \right\}$$
Define $S$ as the sum of the product $...
- 3,684
4
votes
1
answer
301
views
Is this a known result on graph products?
Consider two undirected graphs $G=(V,E)$ and $H=(I,F)$.
Denote by $\mathcal N_G(v)$ (resp., $\mathcal N_{H}(i)$) the first neighborhood of a node $v\in V$ (resp., $i\in I$), including $v$ (resp., $i$)....
- 434
1
vote
1
answer
79
views
Can I factorize a double sum into a product?
Fix two positive constants $A,B>0$, two finite sets $\mathcal A, \mathcal B$, and two functions $\alpha,\beta \colon \mathcal A \times \mathcal B \to [0,1]$.
Assume that:
For all $b\in \mathcal B$,...
- 434