All Questions
22
questions
1
vote
1
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129
views
Proving $\sum_{1}^{n} \left\lceil\log_{2}\frac{2n}{2i-1}\right\rceil=2n -1 $
Show that $$\sum_{i=1}^{n} \left\lceil\log_{2}\frac{2n}{2i-1}\right\rceil=2n -1 $$ where $ \lceil\cdot\rceil$ denotes the ceiling function.
My method: one way would be observe each part of the ...
- 977
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
2
votes
4
answers
62
views
For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$
For a fixed $k$ what is the value of $\sum_{l=1}^{5^m-1} \Big\lfloor \dfrac{l}{5^k}\Big \rfloor$
By dividing the numbers between $1$ and $5^m$ as intervals of $5^k$, I was getting the following ...
user675768
3
votes
2
answers
690
views
How many values of $n$ are there for which $n!$ ends in $1998$ zeros?
How many values of $n$ are there for which $n!$ ends in $1998$ zeros?
My Attempt:
Number of zeros at end of $n!$ is
$$\left\lfloor \frac{n}{5}\right\rfloor+\left\lfloor\frac{n}{5^2}\right\rfloor+\...
- 9,599
2
votes
1
answer
449
views
Sum of all divisors of the first $n$ positive integers.
Yesterday I was going through Möbius Function notes, and found that
writing $n = p_{1}^{\alpha_1}p_{2}^{\alpha_2}\cdots p_{r}^{\alpha_r}$,
the sum of all divisors can be written as.
$$
e(n) = \prod_{...
- 49
3
votes
2
answers
228
views
Find n in sum that results in a number $aaa$
Lets say that we have the sum $1+2+3+\ldots+n$ where $n$ is a positive natural number and that this sum should equal a three digit number in which all the digits are the same, for example $111, 222,$ ...
- 39
0
votes
1
answer
100
views
Is this a valid definition of the rationals?
$$\mathbb{Q}=\left\{\sum_{n=1}^k f(n)\mid k,n\in\mathbb{N}\land f\text{ is a finite composition of $+$, $-$, $\div$, $\times$}\right\}$$
Reasoning:
Any real number can be described by a (sometimes ...
- 5,040
3
votes
2
answers
199
views
Calculate $ \biggr\lfloor \frac{1}{4^{\frac{1}{3}}} + \frac{1}{5^{\frac{1}{3}}} + ... + \frac{1}{1000000^{\frac{1}{3}}} \biggr\rfloor$
Calculate $ \biggr\lfloor \frac{1}{4^{\frac{1}{3}}} + \frac{1}{5^{\frac{1}{3}}} + \frac{1}{6^{\frac{1}{3}}} + ... + \frac{1}{1000000^{\frac{1}{3}}} \biggr\rfloor$
I am just clueless. I just ...
- 2,132
2
votes
2
answers
84
views
Suppose that $1+2+...+n=\overline{aaa}$. Which of the following items CERTAINLY divides $n$? $5,6,7,8,11$
Suppose that $1+2+...+n=\overline{aaa}$. Which of the following items CERTAINLY divides $n$?
$5,6,7,8,11$
I converted the given relation into the following:
$$n(n+1)=2*3*37*a$$
Now I think ...
- 3,455
3
votes
2
answers
102
views
How many integers $n$ for $3<n<100$ are such that $1+2+3+\cdots+(n-1)=k^2$, with $k \in \mathbb{N^*}$?
I know that the sum $1+2+3+\cdots+(n-1)$ equals $\frac{(n-1)\cdot n}{2}$.
I wrote the equation in the two following forms:
$$(n-1)\cdot \frac{n}{2}=k^2$$
$$(n-1)\cdot n=2k^2$$
And I tried to find ...
- 593
4
votes
1
answer
503
views
Does anyone know how to reduce this sum of sums into something simpler in order to find a special value? [duplicate]
to clarify the difference between this and the supposed duplicate, these two questions talk about completely different functions with completely different purposes
I was given this from a friend. ...
- 2,175
0
votes
2
answers
124
views
Proof of Number Theoretic Function $\sigma$ [closed]
If $N$ is a positive integer then,$$\sum\limits_{n=1}^{N}\sigma(n)=\sum\limits_{n=1}^{N}n\lfloor\dfrac{N}{n}\rfloor$$, where $\lfloor.\rfloor$ denotes greatest integer function and $\sigma(n)$ denotes ...
- 1,943
3
votes
4
answers
186
views
Summation or Integral representation ${e^{2}\above 1.5pt \ln(2)}=10.66015459\ldots$
How can I construct a summation or integral representation of $${e^2\above 1.5pt \ln(2)}.$$ Naively I would write $$\Bigg(\sum_{n=0}^{\infty}{2^n \above 1.5pt n!} \Bigg)\Bigg(\sum_{n=1}^\infty {(-1)^{...
- 3,758
11
votes
1
answer
307
views
Proving if it is possible to write 1 as the sum of the reciprocals of x odd integers
Let $x$ be an even number. Is it possible to write 1 as the sum of the reciprocals of $x$ odd integers? Write a proof supporting your answer.
I tried a lot of these, and I think it is no because I ...
- 311
14
votes
4
answers
462
views
Show that the numerator of $1+\frac12 +\frac13 +\cdots +\frac1{96}$ is divisible by $97$
Let $\frac{x}{y}=1+\frac12 +\frac13 +\cdots +\frac1{96}$ where $\text{gcd}(x,y)=1$. Show that $97\;|\;x$.
I try adding these together, but seems very long boring and don't think it is the right way ...
- 153
2
votes
5
answers
848
views
$33^{33}$ is the sum of $33$ consecutive odd numbers. Which one is the largest? (Q25 from AMC 2012)
The number $33^{33}$ can be expressed as the sum of $33$ consecutive odd numbers. The largest of these odd numbers is
$\mathrm{A.}\ 33^{32} +32$
$\mathrm{B.}\ 33^{31} +32$
$\mathrm{C.}\ 33^{32} -32$
$\...
- 749
6
votes
1
answer
212
views
Find the remainder when the sum is divided by $1000$
Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$
$$S_0 = 0! + 0 - 0 + 0 -1 = 0$$
$$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$
$$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$
This isn't helping, so:...
- 11.2k
2
votes
0
answers
150
views
Evaluate this product $n \times \frac{n-1}{2} \times \dots \times \frac{n-(2^k-1)}{2^k}$
For $k = \lfloor \log_{2}(n+1) \rfloor - 1$ evaluate
$n \times \frac{n-1}{2} \times\frac{n-3}{4} \times \frac{n-7}{8} \times \dots \times \frac{n-(2^{k}-1)}{2^k}$
So the product goes up to $k$ and I ...
- 69
1
vote
1
answer
127
views
On $\lfloor\sqrt n \rfloor+ \sum_{j=1}^n \lfloor n/j\rfloor$ [duplicate]
How do we prove that $\Big[\sqrt n \Big]+ \sum_{j=1}^n \bigg[ \dfrac nj\bigg]$ is an even integer for all $ n \in \mathbb N$ ? (where $\Big[ \space \Big]$ denotes the "greatest integer" function)
- 8,387
0
votes
1
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417
views
amortized analysis
a) define f(k) as the largest power of 2 that divides k.
For example, f(25) = 1, f(42) = 2, f(144) = 16.
What is ${1 \over k}\sum_1^k f(k)$?
b) define f(k) as the square of largest power of 2 that ...
- 147
9
votes
3
answers
2k
views
Which number was removed from the first $n$ naturals?
A number is removed from the set of integers from $1$ to $n$. Now, the average of remaining numbers turns out to be $40.75$. Which integer was removed?
By some brute force, I got $61$. I want to know ...
- 1,949
1
vote
2
answers
133
views
Intuition for the following change of index of summation
I am working through Concrete Mathematics. I came across the following change in index of summation while going through the number theory chapter.
$$\sum_{m|n}^{ } \sum_{k|m}{ } a_{k,m} = \sum_{k|n}^...
- 140