All Questions
Tagged with elementary-number-theory summation
482
questions
2
votes
1
answer
200
views
A divisor sum involving Moebius function and Jordan's totient function
I am trying to prove the following claim:
Let $\mu(n)$ be the Moebius function and let $J_k(n)$ be the Jordan's totient function. Then,
$$\displaystyle\sum_{d \mid n} \frac{\mu^2(d)}{J(k,d)}=\frac{n^...
- 12.9k
1
vote
1
answer
64
views
A divisor sum involving generalized Moebius function
I am trying to prove the following claim:
Let $n$ be a squarefree natural number. Denote by $\mu_k$ the generalized Moebius function: $\mu_k=\underbrace{\mu \ast \ldots \ast \mu}_{k}$ where $\ast$ is ...
- 12.9k
1
vote
1
answer
158
views
Maximum sum of digits from a given formula
The sum of digits of the telephone number aaabbbb equals the two-digit number ab.
What is the sum $a+b?$
$(A) \space8$
$(B)\space 9 $
$(C) \space10 $
$(D)\space 11$
$(E) \space 12$
So far I have tried ...
- 23
2
votes
4
answers
2k
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Calculate the sum of the digits of the first 100 numbers of that sequence which are divisible by 202.
In the sequence 20, 202, 2020, 20202, 202020, ... each subsequent number is obtained by adding the digit 2 or 0 to the previous number, alternately. Calculate the sum of the digits of the first 100 ...
- 29
1
vote
0
answers
38
views
Non-geometric proof to lemma for quadratic reciprocity
There is a common result that for odd, relatively prime positive integers $a,b,$ $\sum\limits_{x=1}^{\frac{b-1}{2}}\lfloor\frac{ax}{b}\rfloor+\sum\limits^{\frac{a-1}{2}}_{y=1}\lfloor\frac{by}{a}\...
- 41
1
vote
1
answer
183
views
How summation is changed in Analytic number theory
Consider this expression S(x, z) = $\sum_{n\leq x} \sum_{{d|n , d|P(z) } }\mu(d) $ . I don't understand the logic behind next step and get really confused on how summation is changed.
In next step ...
user775699
5
votes
1
answer
232
views
What is the reminder when $1^1+2^2+3^3+\ldots+50^{50}$ is divided by 8
I have a sum that goes as follows $$P=1^1+2^2+3^3+\ldots +49^{49}+50^{50}.$$
The question is to find the reminder when $P$ is divided by 8.
Inorder to find this, I seperated $P$ into $P_1$ and $P_2$ ...
- 159
4
votes
1
answer
226
views
A proof of the theorem $1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ with $k$ odd and $m$ even
I tried to prove the following theorem :
If $n>1$ then $\displaystyle1+\frac{1}{2}+\dots+\frac{1}{n} = \frac{k}{m}$ where $k$ is odd and $m$ is even.
and I'd like to know if there is any flaw in ...
- 3,990
1
vote
0
answers
50
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Is the value of the sum squarefree for every $n\ne 27\ $?
For a positive integer $\ n\ $ , define $$f(n)=|\sum_{j=1}^n (-1)^j\cdot j!|=n!-(n-1)!+(n-2)!-(n-3)!\pm \cdots$$
Is $\ f(n)\ $ squarefree except for $\ n=27\ $ in which case $\ 127^2\ $ is a factor ?
...
- 85.1k
6
votes
1
answer
98
views
Conjecture about prime factors of a special sum
For a positive integer $\ n\ $ , define $$f(n)=|\sum_{j=1}^n (-1)^j\cdot j!|=n!-(n-1)!+(n-2)!-(n-3)!\pm \cdots$$
I want to prove the
Conjecture : Every prime factor $\ p\ $ of $\ f(n)\ $ must satisfy ...
- 85.1k
2
votes
1
answer
77
views
equality involving sums
Let $n\in\mathbb{Z}^+.$ Prove that for $a_{i,j}\in\mathbb{R}$ for $i,j = 1,\cdots, n,$
$$\left(\sum_{i=1}^n \sum_{j=1}^n a_{i,j}\right)^2 + n^2\sum_{i=1}^n\sum_{j=1}^n a_{i,j}^2 - n\sum_{i=1}^n \left(\...
- 1,225
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 ...
1
vote
2
answers
195
views
Prove $\sum_{n \leq x} \tau (n) = 2(\sum_{n \leq \sqrt{x}} [\frac{x}{n}]) - [\sqrt{x}]^2$
Here $\tau(n)$ is the number of positive integers dividing $n$ and $[x]$ is the floor of $x$.
So I know if $n$ is not a perfect square, then half the positive integers dividing $n$ are less than $\...
- 1,178
1
vote
1
answer
122
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Question on Summation Formula for Fermat Quotients
According to Wolfram (https://mathworld.wolfram.com/FermatQuotient.html),
$\displaystyle \frac{2^{p-1}-1}{p}=\frac{1}{2}\sum_{n=1}^{p-1}\frac{\left(-1\right)^{n-1}}{n}$
I am struggling to understand ...
- 175
-1
votes
1
answer
55
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Some alternating sum of integer part of $\frac{kb}{1722}$.
For every integer $k$ coprime to 1722, how can one compute the sum
$\sigma(k)= \sum_{1\leq b\leq 1722, (1,b)=1722} \lfloor\frac{kb}{1722}\rfloor (-1)^{b-\lfloor \frac{b}{2}\rfloor-\lfloor \frac{b}{3}\...
- 119