Questions tagged [integer-sequences]

For questions about sequences of integers. References are often made to the online resource oeis.org.

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3
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1answer
194 views

The sequence $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$

Related to this question. Let $p$ be prime and $n$ positive integer. Define $a(n)=(2^n \bmod p)^{p-1} \bmod p^2$ Let $D(n)$ be the base $2$ discrete logarithm of $a(n)$, i.e. given $p,a(n)$ we have $2^...
5
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1answer
213 views

Discrete logarithm and the sequence $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$

Let $p$ be prime and $g,n$ integers. Define $a(n)=(g^n \bmod p)^{p-1} \bmod p^2$ By mod p we don't mean congruence, but the reduction modulo $p$ operator. $A \bmod ...
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1answer
189 views

Sequences over finite fields

Let's we have finite field $F_q$ for some prime $q=2^M-1$. I am looking for special sequence {$a_{i}$, $i \in {1,..,q-1}$}, ($\{a_{1},...,a_{q-1}\}=F_q/\{0\}$) with the following properties: $r_{1}=...
2
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2answers
238 views

sum of odious numbers to the power of k

In number theory, an odious number is a positive integer that has an odd number of $1$s in its binary expansion. The first odious numbers are: $1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 21, 22, 25, 26, 28, ...
10
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0answers
236 views

Convergence of rivers of numbers

$\DeclareMathOperator{\river}{river}\DeclareMathOperator{\leadingsum}{ls}\DeclareMathOperator{\digitsum}{ds}\newcommand{\qed}{\square} $A 1999 British Informatics Olympiad question asks about ...
5
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0answers
113 views

Upper bounds for a sequence of integers

Given $\alpha\geq0$ we consider the sequence $$ C_k=k^\alpha\sum_{j=0}^{k-1}C_jC_{k-1-j} $$ with $C_0=1$. I'm interested in upper bounds (in terms of $\alpha$) for such a sequence. I know that when $\...
0
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0answers
89 views

Writing integers as sequences of products by 2 and integer divisions by 3

For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$. For instance: $$ 100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
7
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0answers
181 views

Subwords of the infinite Fibonacci word

Let $W = 01001010010010 \ldots$ be the infinite Fibonacci word, A003849 in the OEIS. Let $B(m)$ be the set of $m+1$ subwords of $W$ that have length $m$, and for each such subword $u$, let $p(u)$ be ...
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0answers
36 views

Calculating ordinal numbers for sorted sequences

given a set $\mathcal{S}=\lbrace\,\sigma\in\mathbb{N}_0^k\ \ \big|\ \ i\lt j\implies\sigma\left[i\right]\le\sigma\left[j\right]\,\rbrace$ of fixed size sequences of natural numbers whose entries are ...
5
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1answer
153 views

Is this Laurent phenomenon explained by invariance/periodicity?

In Chapter 4 of his Tracking the Automatic Ant, David Gale discusses three families of recursively defined sequences of numbers, all due to Dana Scott and inspired by the Somos sequences: Sequence 1. ...
2
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1answer
245 views

Why does this “factorial sequence” appear in the OEIS?

For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$ $$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$ I ...
16
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0answers
827 views

The easily bored sequence

If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
17
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2answers
879 views

A conjecture harmonic numbers

I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven). From the Online Encyclopedia of Positive Integers we have: $a(n)$ ...
7
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0answers
130 views

Factor-counting sequence

Define a non-negative integer sequence $\{\mathcal{F}_n\}$ as follows: start with 1 and, at each step, insert the number of entries already present in the sequence which are factors of the last one. ...
3
votes
1answer
122 views

Properties of a certain sequence

During research I came to the following sequence: Let $\lambda>1$ and define $n_{k+1}=\text{IntergerPart}[\lambda\cdot n_k]$ where we assume that $n_0$ is sufficently large integer, so that the ...
11
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1answer
1k views

Biased random Fibonacci sequences

I have recently been toying (very superficially) with the random Fibonacci sequence, i.e., defined by $F_0=1=F_1=1$ and $$ F_{n} = F_{n-1} + \varepsilon_n F_{n-2} $$ where $(\varepsilon_n)_{n\geq 2}$ ...
3
votes
1answer
242 views

Quadratic progressions with very high prime density

In my previous MO question (see here), I solved the case for arithmetic progressions $f_k(x)=q_k x+1$. The solution is this: The list of sequences $f_k(x)$, each one corresponding to a specific $k$, ...
0
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0answers
22 views

Raggedness measure of a sequence

This surely has been done, maybe I googled the wrong adjective... Define a raggedness measure $r$ of a sequence $S$ in this way: Two members $S_i,S_j$ of the sequence (who don't have to be adjacent!) ...
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0answers
97 views

Question related to sequence of recurrence relation $a_k=\operatorname{rad}(a_{k-1}+a_{k-2})$ for $k\ge 2$ where $a_0=0,a_1=1$

Define radical of an integer Wiki $$\displaystyle{\mathrm{rad}}(n)=\prod_{{\scriptstyle p\mid n\atop p\:{\text{prime}}}}p$$ Example $n=504=2^3\cdot3^2\cdot7$ therefore ${\displaystyle \operatorname{...
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0answers
273 views

Why can one compute the sum of divisors of $n$ without factoring $n$?

Question links to paper which states: $$ \sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1) $$ where $\sigma(n)$ is the sum of divisors of $n$. Another similar ...
22
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1answer
1k views

Parity of the multiplicative order of 2 modulo p

Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
0
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1answer
117 views

Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube

The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$ \begin{equation} \label{one} \int_0^1 \...
4
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1answer
291 views

$\pi(x+200)-\pi(x)\leq 50$?

Is it true, that $\forall x \in \mathbb N, \pi(x+200)-\pi(x) \leq 50 $ ? $$\pi(x)=\text{card}(\{n \in [0,x] \cap \mathbb N, n\text{ is prime}\})$$
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0answers
129 views

On smoothness and roughness of a number related to triangular numbers

Define $\triangle_n$ to be the $n$th triangular number. Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$ Define $(\ell,k)$-smough numbers to be numbers that ...
4
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0answers
269 views

On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$

My question is related to https://oeis.org/A269839. It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. ...
8
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0answers
218 views

Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial

This is mostly a reference request. I'm working with complex coefficients, although all I have in mind have integer coefficients. Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
6
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0answers
169 views

On nontotient Fibonacci numbers

This question is related to sequence of numbers $t$ such that $F_{6t}$ is a nontotient where $F_n$ represents the sequence of Fibonacci numbers for $n\geq 0$. The online encyclopedia Wikipedia has the ...
6
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0answers
154 views

Some conjectural congruences involving Domb numbers

The Domb numbers are given by $$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$ Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....
2
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0answers
117 views

A question on $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$

Let $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$ where $\pi(n)$ is the prime-counting function. By definition, it is obvious that $a_1(n) = n$ and $a_2(n)$ is https://oeis....
2
votes
2answers
277 views

Why are attempts to define chaos with discrete states so scarce?

Interestingly, the theory of nested recurrence relations has been correlated with “discrete chaos” by Golomb (1991) and Tanny (1992). And in literature, there are very few studies that have different ...
18
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0answers
711 views

A question related to Hofstadter-Conway \$10000 sequence

Hofstadter-Conway \$10000 sequence is defined by nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is https://oeis.org/A004001 and it is well-known ...
1
vote
1answer
115 views

(Translation request) Hypotheses of the Blom-Fredberg bounds on denumerants?

I don't know Swedish and I'm not finding the article "G. Blom and C. E. Froberg, On money changing" translated into English... so I tried to read the original (Swedish) with the help of ...
6
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1answer
205 views

On the growth and bounds for a certain sequence of integers known as Bogotá numbers

A Bogotá number is a non-negative integer equal to some smaller number, or itself, times its digital product, i.e. the product of its digits. For example, 138 is a Bogotá number because 138 = 23 x (2 ...
6
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0answers
150 views

My research paper involves computing additional terms of an existing OEIS sequence. Should I first amend the sequence or publish the results?

In the course of my research I computed terms of an existing OEIS sequence that are currently unknown. Having prepared my paper for publication, I am now faced with a (small) dilemma: Do I first ...
6
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0answers
160 views

A recursion which defines polynomials with integer coefficients?

Let $[n]=1+q+\dots+q^{n-1}$ and $u(n)=\prod_{j=1}^n \gcd([j],[n])$. Define $$r(n)=\sum_{d|n,d>1}{(-1)^d \frac{u(n)}{du(\frac{n}{d})^d}r\Big(\frac{n}{d}\Big)^d}+\frac{(1-q)^{n-1}u(n)}{n[n]}$$ with $...
4
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1answer
129 views

A binomial coefficient identity involving two parameters

In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$: $$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=...
11
votes
3answers
632 views

Series and sequences in physical systems & closed form expressions

I gave a colloquium a while ago about physics inspiring recent developments in mathematics and as is almost borderline cliche in such talks, I mentioned the Fibonacci sequence with closed form ...
6
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3answers
791 views

What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?

From Terry Tao's post here there is the statement: "Conversely, if one can somehow establish a bound of the form $$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$ ...
2
votes
1answer
148 views

Distinct distances between adjacent equal elements

Let's call a sequence $a_1, \ldots, a_n$ suitable if for any positive integer $d$ there is at most one index $i$ such that $a_i = a_{i + d}$ and all elements $a_{i + 1}, \ldots, a_{i + d - 1}$ are not ...
0
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0answers
44 views

A distribution of maximum of sums if add to the minimal

Consider a vector of $n$ integer variables with initial values of 0. Each step we take random $w_i\thicksim NB(q, l)$ (independent randon values with the same negative binomial distribution) and add ...
4
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0answers
92 views

When is $\lfloor C^n \rfloor \mod b$ efficiently computable?

For real irrational $C > 1 $ and natural $n,b$, define $a(C,n,b)=\lfloor C^n \rfloor \mod b$ Q1 For which $C,b$ is $a(C,n,b)$ computable in time polynomial in $\log{n}$? Searching in OEIS ...
3
votes
1answer
99 views

How many flips of a fair coin are needed to get at least one run of at least $k$ consecutive heads with probability $\ge1/2$?

The following question was asked today: How many flips $n$ of a fair coin are needed to get at least one run of at least $k$ consecutive heads with probability $P_{k,n}\ge1/2$? The question was ...
1
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0answers
50 views

On a type of equations that involve certain multiplicative functions and polynomials, in relation to their number of solutions

Past weekend I was interested in the sequence A058891 from the On-Line Encyclopedia of Integer Sequences, from this, inspired by the equation due to Benoit Cloitre (2002) that shows the comments, I ...
0
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0answers
106 views

Roots of a family of 4-parameter polynomials

Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by $$ f(x)=x^q-kx^{q-p}-\ell. $$ This polynomial is related to a family of two-...
4
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0answers
110 views

Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers

The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$ and $$L_0=2,\ L_1=1,\ \text{...
0
votes
1answer
59 views

Ordered $m$-tuples with fixed number of changes

Given $1\leq k\leq m$, $2\leq d\leq c i\ln i$ and $2\leq i\leq c'\ln(mi\ln i)$ at some $c,c'>0$ how many sequences (lower and upper bounds) are of form $$z_1,\dots,z_m$$ on the condition that $$0\...
15
votes
1answer
662 views

Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern. We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
1
vote
2answers
286 views

A question about integer triples

How can we generate all integer solutions of the equation $$(qr+rp+pq)(x^2+y^2+z^2) = (p^2+q^2+r^2)(yz+zx+xy),$$ given that $p,q,r$ are integers? Clearly if any one of $(x,y,z), (x,z,y), (y,z,x), (...
1
vote
1answer
225 views

Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$?

I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers. Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^...
1
vote
1answer
173 views

Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function

For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...

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