All Questions
37
questions
1
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59
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Changing index of a double summation [duplicate]
How does one prove the following result, where $x$ is a three-parameter function defined on $\mathbb Z^3$? $$ \sum_{\ell=1}^{P}\sum^{\ell-1}_{i=0} x(\ell,i,\ell-i) \quad = \quad \sum^{P}_{j=1}\sum^{P-...
- 1,139
1
vote
1
answer
73
views
Prove or disprove that $\sum_{k=1}^p G(\lambda^k) = ps(p)$
Prove or disprove the following: if $\lambda$ is a pth root of unity not equal to one, $G(x) = (1+x)(1+x^2)\cdots (1+x^p),$ and $s(p)$ is the sum of the coefficients of $x^n$ for $n$ divisible by $p$ ...
- 1,225
4
votes
3
answers
666
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Representing the cube of any natural number as a sum of odd numbers
I'm expanding my notes on exercises from Donald Knuth's The Art of Computer Programming, and found something rarely mentioned in the Internet, but still useful to prove Nicomachus' Theorem about the ...
- 848
2
votes
0
answers
60
views
Closed form for the sum of 2 raised to the proper divisors of a positive integer
Is there a known closed form for the sum of 2 raised to the proper divisors of a positive integer?
The problem arises in counting the number of binary sequences that can be "cycles" that ...
- 21
4
votes
1
answer
301
views
Is this a known result on graph products?
Consider two undirected graphs $G=(V,E)$ and $H=(I,F)$.
Denote by $\mathcal N_G(v)$ (resp., $\mathcal N_{H}(i)$) the first neighborhood of a node $v\in V$ (resp., $i\in I$), including $v$ (resp., $i$)....
- 434
1
vote
1
answer
79
views
Can I factorize a double sum into a product?
Fix two positive constants $A,B>0$, two finite sets $\mathcal A, \mathcal B$, and two functions $\alpha,\beta \colon \mathcal A \times \mathcal B \to [0,1]$.
Assume that:
For all $b\in \mathcal B$,...
- 434
2
votes
1
answer
202
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If $xy = ax + by$, prove the following: $x^ny^n = \sum_{k=1}^{n} {{2n-1-k} \choose {n-1}}(a^nb^{n-k}x^k + a^{n-k}b^ny^k),n>0$
If $xy = ax + by$, prove the following: $$x^ny^n = \sum_{k=1}^{n} {{2n-1-k} \choose {n-1}}(a^nb^{n-k}x^k + a^{n-k}b^ny^k) = S_n$$ for all $n>0$
We'll use induction on $n$ to prove this.
My ...
6
votes
2
answers
379
views
Two sets having the same subset sums.
I was trying to prove the following
Proposition:
Let $A=\{a_1,\ldots, a_k\}$ and $B=\{b_1,\ldots, b_k\}$ be two multisets (repetition is allowed)
with $|A|=|B|=k$. Also $0\le a_1\le a_2\le\ldots \le ...
- 9,173
4
votes
0
answers
90
views
Prove that $\sum_{r=2}^{n} \left \lfloor n^{\frac{1}{r}} \right \rfloor = \sum_{r=2}^{n} \left \lfloor \log_{r}(n) \right \rfloor$.
Prove that
$$\sum_{r=2}^{n} \left \lfloor n^{\frac{1}{r}} \right \rfloor = \sum_{r=2}^{n} \left \lfloor \log_{r}(n) \right \rfloor\,.$$
I have tried to use substitutions of $n=p^k$ in order to try ...
- 331
2
votes
1
answer
63
views
For any prime $p>3$ show that $C_{np}^{p}-C_{np}^{2p}+C_{np}^{3p}-C_{np}^{4p}+...+(-1)^{n-1}C_{np}^{np} \equiv 1\pmod{p^3}$
Let $n$ be a positive integer. For any prime $p>3$ show that
$$C_{np}^{p}-C_{np}^{2p}+C_{np}^{3p}-C_{np}^{4p}+...+(-1)^{n-1}C_{np}^{np} \equiv 1\pmod {p^3}$$ Where $C_{n}^{k}=\frac{n!}{k!(n-k)!}$. (...
- 11.6k
1
vote
0
answers
121
views
Expressing a sum over the sizes of the parts of every partition of n
Let $(a_1^{r_1},\ldots,a_{p}^{r_{p}})\vdash n$ be the multiplicity representation of an integer partition of n. Each $a_{i}$ is a part of the partition and $r_{i}$ is its corresponding size. We ...
- 1,401
0
votes
3
answers
84
views
prove $\sum_{i=1}^{n} \max\left\{k\in\mathbb {N} \left| \frac{i}{2^k}\in\mathbb{N} \right.\right\}=n-1$
I'm trying to prove the following:
$$\sum_{i=1}^{n} \max\left\{k\in\mathbb {N} \left| \frac{i}{2^k}\in\mathbb{N} \right.\right\}=n-1 \\ \text{where}\ n=2^x,\, x\in\mathbb{N}$$
If I write down the ...
- 322
4
votes
1
answer
250
views
Summing $\sum\frac{1}{i}$ with two constraints on $i$
Let's fix a positive integer $n$ and consider two positive composites $a,b \le n$. We can consider the prime factorizations $a=2^{a_2}3^{a_3}\cdots p^{a_p}$ and $b=2^{b_2}3^{b_3}\cdots p^{b_p}$, where ...
- 3,918
0
votes
1
answer
716
views
Sum of all 5 digit numbers using the digits 1,2,3,4,5 at most once.
Question: By using the digits 1, 2, 3, 4 and 5, 5-digit numbers are formed such that every digit is used at most once, then find the sum of all such possible numbers.
Attempt: I found the answer 3,...
- 405
0
votes
1
answer
114
views
How can we simplify this sum over sets expression?
I am counting some structures with specific properties. I would like to simplify the counting expression in order to have a more easily evaluable function:
$\vcenter{\hbox{$\sum\limits_{U\in2^{2^W\...
- 258