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Results tagged with asymptotics
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user 23388
Asymptotic behavior of functions, asymptotic series and related topics
5
votes
0
answers
352
views
What is the sum of the binomial coefficients ${n\choose p}$ over prime numbers?
What is known about the asymptotics, lower and upper bound of the sum of the binomial coefficients
$$
S_n = {n\choose 2} + {n\choose 3} + {n\choose 5} + \cdots + {n\choose p}
$$
where the sum runs over …
- 5,460
2
votes
0
answers
132
views
Average length of consecutive integers which have an increasing number of divisors
Consider the nine consecutive natural numbers starting from $1584614377$.
n = 1584614377 no. of divisors: 2
n = 1584614378 no. of divisors: 4
n = 1584614379 no. of divisors: 8
n = 1584614380 no. of di …
- 5,460
4
votes
1
answer
234
views
Relation between $\pi$, area and the sides of Pythagorean triangles whose hypotenuse is a pr...
Consider all Pythagorean triangles $a^2 + b^2 = p^2$ in which the hypotenuse $p$ is a prime number. Let $h(x) = \sum_{p \le x}p^2$, $a(x) = \sum_{p \le x}ab$ and $r(x) = \sum_{p \le x}(a+b)^2$. Is it …
- 5,460
6
votes
1
answer
289
views
What is the growth rate of the products of binomial coefficients?
Question 1: Are the following empirically observed relationships true
$$
{n \choose 1^a}{n \choose 2^a}{n \choose 3^a}\cdots {n \choose m^a}
\sim \exp\bigg(\frac{2n^{1 + \frac{1}{a}}}{a+3}\bigg)
$$
…
- 5,460
11
votes
1
answer
323
views
Does the mean ratio of the largest prime factor in prime gaps to the lower bound of the gap ...
Posting in MO since this questions has been unanswered in MSE for 3 months.
Let $p_n$ be the $n$-th prime and $q_n$ be largest among all the prime factors of the composite numbers between $p_n$ and $ …
- 5,460