All Questions
48
questions
0
votes
1
answer
58
views
How to define a function that satifies this condition?
I would like to define a function $f(n)$.
It must be such that it should produce the sum of all elements till the nth term of the series mentioned below:
$$2,2,2,3,3,3,4,4,4,4,5,5,5,6,6,6,7,7,7,7,8,8,...
- 73
3
votes
0
answers
59
views
Is it possible to construct a sequence using the first $n$ prime numbers such that each segment has a unique sum?
For example, consider the sequence $2,7,3,5$. The sums of the segments of this sequence are as follows, and they are all unique:
$$2, 2+7, 2+7+3, 2+7+3+5, 7, 7+3, 7+3+5, 3, 3+5, 5$$
Can we generate ...
- 1,451
0
votes
1
answer
39
views
Differences between sums of reciprocals of primes and products thereof.
I am not a number theorist, but I know that number theorists are notorious for estimating sums including primes. So, I have the following question, which I may not have seen addressed, but I am sure ...
- 717
2
votes
0
answers
60
views
Closed form for the sum of 2 raised to the proper divisors of a positive integer
Is there a known closed form for the sum of 2 raised to the proper divisors of a positive integer?
The problem arises in counting the number of binary sequences that can be "cycles" that ...
- 21
2
votes
4
answers
2k
views
Calculate the sum of the digits of the first 100 numbers of that sequence which are divisible by 202.
In the sequence 20, 202, 2020, 20202, 202020, ... each subsequent number is obtained by adding the digit 2 or 0 to the previous number, alternately. Calculate the sum of the digits of the first 100 ...
- 29
1
vote
1
answer
122
views
Question on Summation Formula for Fermat Quotients
According to Wolfram (https://mathworld.wolfram.com/FermatQuotient.html),
$\displaystyle \frac{2^{p-1}-1}{p}=\frac{1}{2}\sum_{n=1}^{p-1}\frac{\left(-1\right)^{n-1}}{n}$
I am struggling to understand ...
- 175
2
votes
1
answer
110
views
If $p$ and $q$ are coprime positive integers s.t. $\frac{p}{q}=\sum_{k=0}^{100}\frac1{3^{2^k}+1}$, what is the smallest prime factor of $p$?
If the sum $$S=\frac14+\frac1{10}+\frac1{82}+\frac1{6562}+\cdots+\frac1{3^{2^{100}}+1}$$
is expressed in the form $\frac pq,$ where $p,q\in\mathbb N$ and $\gcd(p, q) =1.$ Then what is smallest prime ...
3
votes
1
answer
114
views
Sum of reciprocals of perimeters of primitive Pythagorean triples
The primitive Pythagorean triples, in increasing order of perimeters, are: $(3,4,5),(5,12,13),(8,15,17),\dots$
So, the perimeters of these triangles, in increasing order, are: $12,30,40,\dots$
Let $...
- 5,450
1
vote
0
answers
118
views
co-prime perfect power summation
The following link shows 10000 perfect power.
I was wondering how many numbers are there less than $n$ that are perfect power and also co-prime with $n$. i.e. $gcd(n,k) = 1$.
In general I would like ...
- 49
1
vote
0
answers
408
views
Sum of perfect powers of n natural numbers
The perfect power of an integer $n = p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_r^{\alpha_r}$ are $p_1^2, p_1^3\cdots p_1^{\alpha_1},p_2^2, p_2^3 \cdots p_2^{\alpha_2},(p_1 p_2)^2, (p_2 p_3)^2\cdots$ ...
- 49
1
vote
2
answers
87
views
If $S_{100} - k + k² = 7500$ and $S_{100}$ it is the sum of $100$ consecutive positive integers, then what it is the value o k?
$k$ must be one of the 100 consecutive integers.
I know the answers are $50$ and $26$.
But I got stuck into getting these numbers.
Here is what I tried:
$S_{100} = \frac{(n+n+99)\times100}{2} \...
- 367
1
vote
1
answer
314
views
Sum of arithmetic-geometric progression
I'm looking for a function that generates the sum of the products of the $i$-terms of a geometric and arithmetic progressions. In other words, I'm searching a closed form for the expression: $$\...
- 6,581
4
votes
1
answer
260
views
Role of binomial coefficents in nested summations in layman terms
I has this doubt from almost two years and not getting a simple solution in layman terms. In short, the doubt is the about relation between binomial coefficients and the nesting summation.
Recently I ...
- 196
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
7
votes
2
answers
175
views
Why do expressions of the form $\sum\limits^\infty_{n=1} \frac{n^k}{3^n}$ sum 'nicely'?
Why is it that sums like $\displaystyle\sum\limits^\infty_{n=1} \frac{n^k}{3^n}$ and $\displaystyle\sum\limits^\infty_{n=1} \frac{n^k}{2^n}$ where $k$ is a non-negative integer, sum to 'nice' numbers ...
- 353