All Questions
18 questions
54
votes
2
answers
8k
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
August Ferdinand Möbius (November 17, 1790 – ...
13
votes
1
answer
2k
views
Erdős multiplication problem revisited
This is a well-known problem and is about counting the number of distinct numbers in the $n \times n$ multiplication table.
The very problem has been discussed in-depth and, as such, I require no ...
user56218
11
votes
2
answers
410
views
Extension of Dickson's theorem on integers of the form $a^2+b^2+2c^2$
Theorems V in this paper of L.E. Dickson states that the following two sets are equal. $$E=\{a^2+b^2+2c^2 \ | \ a,b,c \in \mathbb{Z}\} \ \text{ and } \ F=\mathbb{N} \setminus \{4^k(16n+14) \ | \ k,n \...
11
votes
1
answer
278
views
4-cliques of pythagorean triples graph and its connectivity
Let natural numbers $a, b > 2$ be adjacent if $|a^2 - b^2|$ is a square number. One can find a 3-clique.
For example 153, 185, 697. The questions are: does there exist a 4-clique? Is this graph ...
- 333
10
votes
3
answers
1k
views
Explicit formula for elementary symmetric sum
For $k\ge1$, $j\ge1$, Let $$e_k(j)=\sum_{1\le i_1<...<i_k\le j}i_1\cdot\cdot\cdot i_k.$$ We know that $e_k(j)$ is a polynomial in $j$ with coefficients depending on $k$. I am curious about ...
- 113
8
votes
1
answer
484
views
simple conjecture on palindromes in base 10 [closed]
The conjecture says that for any a, b belong to the the set of non-negative integers ($a$ and $b$ are not necessarily distinct), taking any natural value of $c$; we have always that $$(10^c-1) \cdot \...
7
votes
0
answers
294
views
On the ratio of Gilbreath sequences
Definitions: let $n \in \mathbb{N}_{>0} \cup \{ \infty \}$ and let $E_n$ be the set of sequences $(d_i)_{i=1}^n$ such that $d_1=1$, $d_i$ is an even integer (for $i > 1$) and $0<d_i \le i$. ...
6
votes
4
answers
552
views
(Non)uniqueness of the common-factor graph
Let $S=\{x_1,\ldots,x_k\}$ be a set of $k$ distinct natural numbers,
a subset of $\{1,\ldots,n\} = \mathbb{N}_{\le n}$.
Define the common-factor graph $G(S)$ as the (undirected) graph with
a node for ...
- 151k
6
votes
2
answers
385
views
Is the nth-power-sum graph connected?
This post was inspired by the Square-Sum Problem presented in Numberphile by Matt Parker.
He asked about Hamiltonianness for $n=2$, and we ask about connectedness for all $n \in \mathbb{N}^*$.
...
6
votes
2
answers
461
views
Divisibility labeling on a boolean lattice and positive Euler totient
Let $B_n$ be the rank $n$ boolean lattice (i.e. the subset lattice of $\{1,2, \dots , n \}$). Let $\hat{0}$ and $\hat{1}$ be the minimum and the maximum of $B_n$. Let $f: B_n \to \mathbb{N}$ be a ...
5
votes
1
answer
610
views
Eulerian ordering of the integers modulo n
Let $n>1$ be an integer. Consider the set $C_n := \{0,1, \dots , n-1\}$.
An Eulerian ordering of $C_n$ is an ordering $r_1, \dots, r_n$ of its elements such that:
$$\forall i \le n \ \forall j&...
5
votes
0
answers
313
views
A question on infinite arithmetic progressions
I was working on a problem that consisted of deciding if the language a finite automaton (the alphabet of which is $\{0,1\}$ and the words accepted are binary encoded positive integers) contains an ...
- 51
4
votes
1
answer
755
views
Conjecture on palindromic numbers
The conjecture is as follows:
Let $n\in\mathbb{N}\setminus\{1\}$. Define $a(n)=2^n+1$ and the set:
$$S(n) = \{ (a(n)^m+1)/2\ :\ m\in \mathbb{N}_0\}.$$
Then for all $c\in\mathbb{N}$, the number $(a(n)...
3
votes
2
answers
2k
views
Integer partition and sum of squares
Hello,
The question below might be well known, and using different words (I made these up, I'm not a number theorist or specialist in combinatorics)
For all integers $n\geq 2$ denote by $\mathcal{P}...
- 2,829
2
votes
1
answer
199
views
Decide if a system of arithmetic sequences is an $m$-cover of $\mathbb{N}$
Let $A = \{ a_i + b_i \mathbb{N} \}_{i=1}^{k}$, where $a_1, \ldots, a_k \in \mathbb{N} \cup \{0\}$ and $b_1, \ldots, b_k \in \mathbb{N}$ be a system of arithmetic sequences.
For a positive integer $m$...
- 655
1
vote
1
answer
353
views
Valid Difference Sets
Suppose
$$
P \subseteq \{1,2,\dots,N\},\quad |P| = K
$$
We calculate the differences as:
$$
d=p_i-p_j\mod N,\quad i\ne j
$$
Now let $a_d$ denote the number of occurrence of $d$ (for $d = 0, 1, 2, ...
- 123
0
votes
2
answers
3k
views
Fibonacci Numbers Modulo m [closed]
In the paper "Fibonacci Series Modulo m" by D.D. Wall (found here), there is a table in the Appendix listing values for the function $k(p)$. This function is defined as the period of the Fibonacci ...
- 133
0
votes
1
answer
97
views
On polynomials associated to integers power sums [closed]
For $0\leq k\leq n$ integers let $P_k(n):= n^k,\ S_k(n):= P_k(1)+\ldots P_k(n)= 1^k+\ldots n^k$.
Then $P_k(0)=0$, $S_0(n)=n$.
For calculate $S_1(n)$ i consider:
$$P_2(n)-P_2(n-1)=2n+1$$
then
$\begin{...
- 4,165