Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
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Polynomials, $3^x$ and the Collatz conjecture
$\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open.
Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having ...
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Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?
The Collatz or the $3n+1$ conjecture is open.
Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
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Why do we need to represent integers as the sum of three cubes? [closed]
It is conjectured that for any integer $k\not\equiv \pm 4\pmod 9$ there are infinitely many integer solutions to
$$
a^3+b^3+c^3=k.
$$
Some cases for integer $k$ becomes too hard like $42$ which it ...
3
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2
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197
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Limit of the Schröder numbers ratio
I have been playing around with interesting integer sequences and came across Schröder number which defines the number of lattice paths of n x n grid.
The recurrence formula to calculate these numbers ...
22
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Reference request: a tale of two mathematicians
I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so:
When P. Gabriel presented the theorem in a conference [sometime around ...
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1
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The largest digital sum of the square of an n-digit number
The sequence $13, 31, 46, 63, 81, 97, 112, 130, 148, 162, 180, \dots,$ (sequence A348300 in the OEIS) shows the largest digital sum the square of an $n$-digit (decimal) number has.
Is this sequence ...
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Why do convoluted convolved Fibonacci numbers pop up from this triangle?
Start with this triangle (OEIS A118981). This triangle is simple to generate with the following recurrence relation (though $T(0,0)$ ends up different from the OEIS version):
$$
T(0,0) = 2;T(1,0) = 1;...
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On gaps in a sequence of integers
Given a fixed $p \in \{3,4,5,\ldots\}$, we define the strictly increasing sequence $\{a_k\}_{k\in \mathbb N}$ as follows. We set $a_{p,1}=1$ and for each $k>1$, we set $a_{p,k}$ to be the least ...
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Integer sequences with unique $k$-subsets sum
let the $\binom{\mathfrak{M}}{k}$ be a shorthand notation for chosing $k$ elements of set $\mathfrak{M}$ of positive integers and let $\left|\binom{\mathfrak{M}}{k}\right|$ denote the sum of the ...
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Tangent numbers, secant numbers and permanent of matrices
Inspired by Question 402572, I consider the permanent of matrices
$$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$
where $n$ ...
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Where is the source of the formula $\sum_{j=0}^\infty \bigl(j+\frac{1}{2}\bigr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}$ for an integer sequence?
The infinite series representation
\begin{equation}
\frac1\pi\sum_{j=0}^\infty \biggl(j+\frac{1}{2}\biggr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}, \quad n\ge0
\end{equation}
for the positive ...
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Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?)
I think that I might have spotted I small mistake (a missing $5$-defective Lehmer pair) in the classification of terms of Lehmer sequences without primitive divisors given in:
1 Bilu, Hanrot, and ...
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New experiments involving Ramanujan primes: Benford's law
I know that in the literature there are interesting articles involving the sequence of Ramanujan primes, I refer the Ramanujan Prime from the online encyclopedia Wolfram MathWorld. This week I ...
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Bounds for $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$
This is related to problem in graph theory.
OEIS defines A033485 as
$a(1)=1$ and $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$.
Q1 what are upper bounds and asymptotics for $a(n)$, can we get $\exp(o(n))$?
...
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Bounds for the sequence $a(n,A)=n*a(\lfloor (1-A)n \rfloor,A)$
Related to this question and possibly the open problem
of the exponential time hypotheses.
Let $A$ be rational number, $0 < A < 1$.
For positive integer $n$, define the sequence
$a(1,A)=1$ and $(...
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Reference request: Counting integer sequences in homogeneous linear recurrences
Are there references in the literature that deal with the probability of finding an integer sequence in a linear homogeneous recurrence with constant coefficients $ \in \mathbb{Z}$? (or provides a way ...
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Integrality of a sequence formed by sums
Consider the following sequence defined as a sum
$$a_n=\sum_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$
QUESTION. For $n\geq1$, is the sequence of rational numbers $a_n$ ...
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Upper bound an integral with exponential function
I am working on my research about approximation a function. I come up with the following integral. I run some simulations and saw that the integral would converge to zero as n goes to infinty. Here is ...
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Does $a_{i}(n)$ hit every positive integers infinitely many times for all $i\ge1$?
This question is related to a family of sequences. I have a simple definition as below and I have a question based on my limited observations for $i\le200$ and $n \le 10^{9}$.
Definition. $a_{i}(1) = ...
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Avoiding equality of partial sums of two different aperiodic sequences
Consider two distinct sequences of positive integers, $a_{n}|_{n=1}^{\infty}$, and $b_{n}|_{n=1}^{\infty}$ such that for either sequence no period exists. The elements of both sequences are drawn from ...
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220
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Euler quotients modulo $n$
For odd integer $n$, define the Euler quotient modulo $n$ to be $a(n)$:
$$ a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$
$a(n)=0$ for OEIS sequence Wieferich numbers
...
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In the Oldenburger-Kolakoski sequence, is #1s = #2s infinitely many times?
The Oldenburger-Kolakoski sequence, $OK$, is the unique sequence of $1$s and $2$s that starts with $1$ and is its own runlength sequence:
$$OK = (1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,\ldots).$...
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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^...
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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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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}=...
- 33
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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, ...
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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 ...
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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 $\...
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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:
$$
100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
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1
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Subwords of the infinite Fibonacci word
Let $W = 01001010010010 \ldots$ be the infinite Fibonacci word, A003849 in the OEIS. Let $B(m)$ be the set of $m+1$ subwords of $W$ that have length $m$, and for each such subword $u$, let $p(u)$ be ...
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Is this Laurent phenomenon explained by invariance/periodicity?
In Chapter 4 (page 23, subsection "Somos sequence update") of his Tracking the Automatic
Ant, David Gale
discusses three families of recursively defined sequences of numbers, all
due to Dana ...
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5
votes
1
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Why does this "factorial sequence" appear in the OEIS?
For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$
I ...
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The easily bored sequence
If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
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A conjecture harmonic numbers
I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven).
From the Online Encyclopedia of Positive Integers we have:
$a(n)$ ...
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Factor-counting sequence
Define a non-negative integer sequence $\{\mathcal{F}_n\}$ as follows: start with 1 and, at each step, insert the number of entries already present in the sequence which are factors of the last one.
...
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Properties of a certain sequence
During research I came to the following sequence:
Let $\lambda>1$
and define $n_{k+1}=\text{IntergerPart}[\lambda\cdot n_k]$ where we assume that $n_0$ is sufficently large integer, so that the ...
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1
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Biased random Fibonacci sequences
I have recently been toying (very superficially) with the random Fibonacci sequence, i.e., defined by $F_0=1=F_1=1$ and
$$
F_{n} = F_{n-1} + \varepsilon_n F_{n-2}
$$
where $(\varepsilon_n)_{n\geq 2}$ ...
- 1,372
4
votes
1
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435
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Quadratic progressions with very high prime density
In my previous MO question (see here), I solved the case for arithmetic progressions $f_k(x)=q_k x+1$. The solution is this:
The list of sequences $f_k(x)$, each one corresponding to a specific
$k$, ...
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0
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Raggedness measure of a sequence
This surely has been done, maybe I googled the wrong adjective...
Define a raggedness measure $r$ of a sequence $S$ in this way:
Two members $S_i,S_j$ of the sequence (who don't have to be adjacent!) ...
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Question related to sequence of recurrence relation $a_k=\operatorname{rad}(a_{k-1}+a_{k-2})$ for $k\ge 2$ where $a_0=0,a_1=1$
Define radical of an integer Wiki
$$\displaystyle{\mathrm{rad}}(n)=\prod_{{\scriptstyle p\mid n\atop p\:{\text{prime}}}}p$$
Example $n=504=2^3\cdot3^2\cdot7$ therefore ${\displaystyle \operatorname{...
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Why can one compute the sum of divisors of $n$ without factoring $n$?
Question links to paper
which states:
$$
\sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1)
$$
where $\sigma(n)$ is the sum of divisors of $n$.
Another similar ...
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24
votes
1
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Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
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1
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144
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Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube
The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$
\begin{equation} \label{one}
\int_0^1 \...
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$\pi(x+200)-\pi(x)\leq 50$?
Is it true, that $\forall x \in \mathbb N, \pi(x+200)-\pi(x) \leq 50 $ ?
$$\pi(x)=\text{card}(\{n \in [0,x] \cap \mathbb N, n\text{ is prime}\})$$
- 4,074
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0
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151
views
On smoothness and roughness of a number related to triangular numbers
Define $\triangle_n$ to be the $n$th triangular number.
Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$
Define $(\ell,k)$-smough numbers to be numbers that ...
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0
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300
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On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$
My question is related to https://oeis.org/A269839.
It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. ...
- 701
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Is there a positive odd $n$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$?
Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203)
It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) ...
- 701
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0
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237
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Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial
This is mostly a reference request.
I'm working with complex coefficients, although all I have in mind have integer coefficients.
Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
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1
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286
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On nontotient Fibonacci numbers
This question is related to sequence of numbers $t$ such that $F_{6t}$ is a nontotient where $F_n$ represents the sequence of Fibonacci numbers for $n\geq 0$.
The online encyclopedia Wikipedia has the ...
- 701
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0
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184
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Some conjectural congruences involving Domb numbers
The Domb numbers are given by
$$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....
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