All Questions
40
questions
2
votes
0
answers
111
views
On $s$-additive sequences
For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
- 9,534
0
votes
0
answers
76
views
Let $f$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$. Is $f(k)$ decreasing for all large $k$?
Consider the question
Let $f:\mathbb N \to \mathbb R$ satisfy $f(mr)<f(r)$ for all $m,r\in \mathbb N$, $m>1$. Is $f(k)$ decreasing for all $k>k_0$ for some $k_0$?
The answer is clearly no ...
1
vote
0
answers
64
views
A question on generalized bases
I just came to know that it is possible to define a generalized base as an infinite sequence of natural numbers $\mathbf b=(b_1,b_2,\dots)$ where $b_i\ge 2$ for all $i$. With this definition, any $m\...
2
votes
1
answer
61
views
Counting gap sizes in a subfamily of partitions
Let $\mathcal{OD}$ be the set of all odd and distinct integer partitions. This has a generating function given by
$$\sum_{\lambda\vdash\mathcal{OD}}q^{\vert\lambda\vert}=\prod_{j\geq1}(1+q^{2j-1})$$
...
- 339
3
votes
1
answer
189
views
On the Finite Sum of Reciprocal Fibonacci Sequences
I want to check if $$\left \lfloor \left( \sum_{k=n}^{2n}{\dfrac{1}{F_{2k}}} \right) ^{-1} \right\rfloor =F_{2n-1}~~(n\ge 3)\qquad(*)$$ where $\lfloor x \rfloor$ is a floor function.
Fibonacci ...
- 1,161
0
votes
0
answers
65
views
Axiomatic books for combinatorics and number theory
I would like to know if there are any books on combinatorics and number theory that follow an axiomatic approach akin to that of Sierpinski's General Topology. I have found some books for both ...
- 1
16
votes
1
answer
985
views
Attempting to restate the question of whether the collatz conjecture has a nontrivial cycle as a combinatorics problem
It occurs to me that the question about whether non-trivial cycles exist for the collatz conjecture can be restated as these two questions (details on how this relates to the collatz conjecture can be ...
- 10.5k
2
votes
2
answers
210
views
Given the GCD and LCM of n positive integers, how many solutions are there?
Question: Suppose you know $G:=\gcd$ (greatest common divisor) and $L:=\text{lcm}$ (least common multiple) of $n$ positive integers; how many solution sets exist?
In the case of $n = 2$, one finds ...
- 14.8k
2
votes
0
answers
110
views
Reference Request: Generalized Limit Definition
I've been working on a framework for dealing with large and small sets. A large set of positive integers is one which the sum of the reciprocals of the members of the set diverges and a small set of ...
- 891
7
votes
2
answers
197
views
Are there any non-casework proofs of the $18$-point problem?
There is a cute little problem stated as follows:
Choose a sequence $x_1,x_2,x_3,\ldots$ with $x_i\in[0,1)$ such that $x_1$ and $x_2$ are in different halves of the unit interval, $x_1,x_2,$ and $x_3$...
- 17.6k
0
votes
1
answer
79
views
Combinatorics and least common multiple
Would you be so kind as to provide me with a hint for a question that I can't solve?
It is supposed to be more or less easy, but I don't see what the quick way to settle is. Let me thank you in ...
- 577
0
votes
1
answer
85
views
Literature on bounds of Fubini's numbers
If anybody can suggest where I can find a literature for a known upper and lower bounds on Fubini numbers https://en.wikipedia.org/wiki/Ordered_Bell_number
- 301
1
vote
1
answer
179
views
furstenberg's proof of szemeredi's theorem
I have been struggling for a long time to understand the ergodic theoretical proof of szemeredi's theorem.
What is the highly recommended reference for furstenberg's proof of szemeredi's theorem ?
- 2,498
5
votes
1
answer
218
views
Pretty $p^2$-congruences involving Stirling numbers of the both kinds
Let $p$ an odd prime number and ${n\brack {k}}$ (resp. ${n\brace k}$) be the Stirling numbers of first (resp. second) kind, such that:
$$ \sum_{k\ge0} {{n}\brack {k}}x^k = \prod_{j=0}^{n-1}(x+j)$$
$$ ...
- 3,716
8
votes
2
answers
168
views
Is there any elementary solution for this problem on colored interval?
The problem is as following.
Assume $m,n$ are two coprime odd numbers, consider the interval $[0,mn]$. We cut the interval by $m,2m,\ldots,(n-1)m$ and $n, 2n,\ldots, (m-1)n$ into $m+n-1$ pieces of ...
- 1,582