4
votes
1answer
229 views

Existence of a certain subset of natural numbers equidistributed modulo $m$ for every $m$

I was talking to a friend and the following set $S$ came up. Let $f$ be some real valued function tending to infinity. Let $S$ be a subset of natural numbers such that $|S \cap [1,N]| = N^{\delta}+ ...
3
votes
2answers
380 views

summation of products of combinatorials

For any natural number $N$ and $0\le n\le N$ define $$ f(n) = f(n,N) = \frac{1}{(N+1)!} \sum_{\substack{{S\subset \{1,\ldots,N\}} \\ {|S|=n}}} \prod_{s\in S} s. $$ (The empty product is interpreted ...
20
votes
3answers
2k views

How many different numbers can be obtained as product of first $n$ natural numbers?

Let m and n be natural numbers, and consider the set of all possible products of m (not necessarily distinct) elements from the set $\{1,2,\ldots,n\}$, that is consider the set $\{1^{a_1} \cdot ...
8
votes
3answers
359 views

On the vanishing of the generalized von Mangoldt function $\Lambda_k(n)$ when $n$ has more than $k$ prime factors

It is a well-known fact that the generalized von Mangoldt function, defined by $$\displaystyle \Lambda_k(n) = \sum_{d | n} \mu(d) \left(\log \frac{n}{d}\right)^k$$ vanishes whenever $n$ has more ...
0
votes
1answer
340 views

How to do such a partitioning?

Assume: $$ P \subseteq \{1,2,\dots,N\},\quad |P| = K, \qquad x \in \mathbb{R}_+^K , \qquad w = e^{-j\frac{2\pi}N} $$ and, $$ f(l) = \sum_{i=1}^K \sum_{j=1}^K x_i x_j w^{(p_i-p_j)l} $$ I am going to ...
0
votes
0answers
131 views

shortest relation between poly-Bernoulli numbers and Euler numbers

Poly-Bernoulli numbers which introduced by M.Kaneko are $B_k^{(n)}$ which satisfies in generating function ${Li_{k}(1-e^{-x}) \over 1-e^{-x}}=\sum_{n=0}^{\infty}B_{n}^{(k)}{x^{n}\over n!}$ where Li ...
1
vote
0answers
149 views

Estimates for the size of the product set [n].[n] [duplicate]

Possible Duplicate: Number of elements in the set {1,…,n}*{1,..,n} Writing $[n]$ for the set $\lbrace1,2,...,n\rbrace$, let $P_n$ denote the product set $[n].[n]$, i.e. $$ P_n = ...
6
votes
2answers
234 views

Minimal period of arithmetic progressions occurring in sets of positive density.

Let $A$ be a subset of ${\mathbb N}$ with positive upper-Banach density, and for each integer $k\geq3$, define $R_k=R_k(A)$ to be the smallest positive integer $r$ such that $A$ contains a length $k$ ...
20
votes
2answers
1k views

Number of elements in the set $\{1,\cdots,n\}\times\{1,\cdots,n\}$

Let $A_n=\{a\cdot b : a,b \in \mathbb{N}, a,b\leq n\}$. Are there any estimates for $|A_n|$? Will it be $o(n^2)$?
17
votes
1answer
912 views

Number of distinct values taken by $\alpha$ ^ $\alpha$ ^ $\dots$ ^ $\alpha$ with parentheses inserted in all possible ways, $\alpha\in\mathbf{Ord}$

Let $\alpha\in\mathbf{Ord}$ and $n\in\mathbb{N}^+$. Let $F_\alpha(n)$ be the number of distinct values taken by ordinal exponentiation $\underbrace{\alpha \hat{\phantom{\hat{}}} \alpha ...
17
votes
1answer
2k views

Möbius Randomness of the Rudin-Shapiro Sequence

The Rudin-Shapiro sequence (also known as the Golay-Rudin-Shapiro sequence) is defined as follows. Let $a_n = \sum \epsilon_i\epsilon_{i+1}$ where $\epsilon_1,\epsilon_2,\dots$ are the digits in the ...
3
votes
1answer
707 views

Jacobsthal function related to squares

The ordinary Jacobsthal function $j$ is defined by setting $j(n)$ as the smallest number $m$, such that for each consecutive $m$ numbers in the integers, there is at least one of the numbers comprime ...
4
votes
1answer
352 views

A question concerning products of finite cyclic groups

Let $m_1,\ldots, m_n$ be pairwise coprime natural numbers $\geq 1$. We consider the product $$G(m_1,\ldots,m_n) := \prod_{i=1}^{n} \mathbb{Z} / m_i \mathbb{Z}.$$ We define $M(n)$ as the set $n$-tuple ...
37
votes
2answers
5k views

Walsh Fourier Transform of the Möbius function

This question is related to this previous question where I asked about ordinary Fourier coefficients. Special case: is Möbius nearly Orthogonal to Morse ! Harold Calvin Marston Morse (24 March ...
14
votes
4answers
4k views

Exact formulas for the partition function?

I am curious, what kind of exact formulas exist for the partition function $p(n)$? I seem to remember an exact formula along the lines $p(n) = \sum_k f(n, k)$, where $f(n, k)$ was some extremely ...
6
votes
3answers
1k views

What is the relationship between the Bell numbers, the Bell polynomials, and the partition numbers?

A friend of mine and I were wondering what relationship exists between the partition numbers $p_{n}$ and the Bell numbers $B_{n}$ (and also possibly the Bell polynomials ...
29
votes
1answer
2k views

Is the set of primes “translation-finite”?

The definition in the title probably needs explaining. I should say that the question itself was an idea I had for someone else's undergraduate research project, but we decided early on it would be ...