All Questions
Tagged with elementary-number-theory summation
482
questions
50
votes
4
answers
5k
views
Identity involving Euler's totient function: $\sum \limits_{k=1}^n \left\lfloor \frac{n}{k} \right\rfloor \varphi(k) = \frac{n(n+1)}{2}$
Let $\varphi(n)$ be Euler's totient function, the number of positive integers less than or equal to $n$ and relatively prime to $n$.
Challenge: Prove
$$\sum_{k=1}^n \left\lfloor \frac{n}{k} \right\...
- 55.8k
44
votes
2
answers
207k
views
How to get to the formula for the sum of squares of first n numbers? [duplicate]
Possible Duplicate:
How do I come up with a function to count a pyramid of apples?
Proof that $\sum\limits_{k=1}^nk^2 = \frac{n(n+1)(2n+1)}{6}$?
Finite Sum of Power?
I know that the sum of ...
- 1,588
36
votes
9
answers
4k
views
Is difference of two consecutive sums of consecutive integers (of the same length) always square?
I am an amateur who has been pondering the following question. If there is a name for this or more information about anyone who has postulated this before, I would be interested about reading up on it....
- 379
35
votes
8
answers
7k
views
Is there a direct, elementary proof of $n = \sum_{k|n} \phi(k)$?
If $k$ is a positive natural number then $\phi(k)$ denotes the number of natural numbers less than $k$ which are prime to $k$. I have seen proofs that $n = \sum_{k|n} \phi(k)$ which basically ...
- 2,077
29
votes
0
answers
1k
views
Are there unique solutions for $n=\sum_\limits{j=1}^{g(k)} a_j^k$?
Edward Waring, asks whether for every natural number $n$ there exists an associated positive
integer s such that every natural number is the sum of at most $s$ $k$th powers of natural
numbers (for ...
- 18.6k
26
votes
1
answer
908
views
Mysterious sum equal to $\frac{7(p^2-1)}{24}$ where $p \equiv 1 \pmod{4}$
Consider a prime number $p \equiv 1 \pmod{4}$ and $n_p$ denotes the remainder of $n$ upon division by $p$. Let $A_p=\{ a \in [[0,p]] \mid {(a+1)^2}_p<{a^2}_p\}$.
I Conjecture
$$\sum_{a \in A_p } a=\...
- 3,792
26
votes
3
answers
1k
views
Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes
Starting from 2, the sequence of sums of all consecutive primes is:
$$\begin{array}{lcl}2 &=& 2\\
2+3 &=& 5 \\
2+3+5 &=& 10 \\
2+3+5+7 &=& 17 \\
2+3+5+...
- 2,827
25
votes
6
answers
10k
views
What is the sum of all positive even divisors of 1000?
I know similar questions and answers have been posted here, but I don't understand the answers. Can anyone show me how to solve this problem in a simple way? This is a math problem for 8th grade ...
- 1,749
24
votes
2
answers
724
views
Every natural number is covered by consecutive numbers that sum to a prime power.
Conjecture. For every natural number $n \in \Bbb{N}$, there exists a finite set of consecutive numbers $C\subset \Bbb{N}$ containing $n$ such that $\sum\limits_{c\in C} c$ is a prime power.
A list of ...
22
votes
14
answers
13k
views
Can the factorial function be written as a sum?
I know of the sum of the natural logarithms of the factors of n! , but would like to know if any others exist.
- 5,630
22
votes
5
answers
2k
views
Show that this sum is an integer.
I have to show that
$$g\left(\frac{1}{2015}\right) + g\left(\frac{2}{2015}\right) +\cdots + g\left(\frac{2014}{2015}\right) $$
is an integer. Here $g(t)=\dfrac{3^t}{3^t+3^{1/2}}$.
I tried to solve ...
- 1,156
21
votes
3
answers
2k
views
Efficient computation of $\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$
I realize that there is probably not a closed form, but is there an efficient way to calculate the following expression?
$$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$
I've noticed $$\sum_{k=...
- 4,814
17
votes
1
answer
10k
views
Sum of GCD(k,n)
I want to find this
$$ \sum_{k=1}^n \gcd(k,n)$$
but I don't know how to solve. Does anybody can help me to finding this problem.
Thanks.
- 333
17
votes
1
answer
322
views
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 $9$
$$9\left(\frac{1}{9}+\frac{1}{99}+\frac{1}{999}+\frac{1}{...
- 1,748
16
votes
4
answers
17k
views
Induction Proof that $x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1})$
This question is from [Number Theory George E. Andrews 1-1 #3].
Prove that $$x^n-y^n=(x-y)(x^{n-1}+x^{n-2}y+\ldots+xy^{n-2}+y^{n-1}).$$
This problem is driving me crazy.
$$x^n-y^n = (x-y)(x^{n-1}+...
- 777