# Questions tagged [integer-sequences]

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

265
questions

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### Why are these Littlewood-Richardson coefficients congruent to 1 mod 8?

Let $n\in{\mathbb N}$ and write $n=q_1+q_2+\dots+q_t$, where $q_1>q_2>\dots>q_t$ are powers of $2$. Let $\lambda_n$ be the partition with Frobenius symbol $(q_1-1,q_2-1,\dots,q_t-1;q_t,q_{t-1}...

2
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1
answer

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### Conjectural congruences for numbers related to Littlewood-Richardson coefficients

For $n \geq 0$, let $a_n$ be the square of the Euclidean length of the vector of Littlewood-Richardson coefficients of $\sum_{\lambda \vdash n} s_\lambda^2$, where $s_\lambda$ are the symmetric Schur ...

6
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0
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### Do you recognize these numbers related to the higher Airy equations?

I'm studying the higher Airy equations
$$\left[\big({-}\tfrac{\partial}{\partial y}\big)^{n-1} - y\right] \psi = 0$$
under a coordinate transformation. The interesting coefficients $c_n^{(1)}, \ldots, ...

2
votes

1
answer

123
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### Difference sequences of sets of integers

In this paper, the conception of the difference sequence and $\infty$-difference length of a subset of groups is introduced. As an important case, subsets of the additive group of integers are ...

1
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0
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### A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...

0
votes

1
answer

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### Reducing recurrence relations mod10 [closed]

I have been playing around with integer sequences as of late, and the following question occurred to me:
Suppose for $m$ fixed we have some some initial values $a_1,\cdots,a_m$ and for all $n\in\...

1
vote

1
answer

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### Invariants ("checksums", "hash") for collection of integers

The sum of a collection of integers doesn't depend on the order of the integers and can detect the corruption of one element of the collection (but multiple elements can get corrupted without their ...

20
votes

2
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552
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### What is the smallest size of a shape in which all fixed $n$-polyominos can fit?

Let $n$ be an integer and consider all fixed $n$-polyominos, i.e., without rotation or reflection. I am interested in finding a shape in which all polyominos can embed. (It is OK if multiple ...

5
votes

1
answer

265
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### Combinatorics related plane geometry

There are $n$ men, standing one at each vertex of a convex $n$-gon. If they are allowed to move together along sides or diagonals of the polygon to reach another vertex, how many different ways are ...

3
votes

1
answer

124
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### Limit associated with two Beatty sequences that are not a Beatty pair

Suppose that $r>1$ and $s>1$ are irrational numbers, and let $a_n=\lfloor nr \rfloor$ and $b_n=\lfloor ns \rfloor$. Assume that $r$ and $s$ are numbers for which $\{a_n\}\cap\{b_n\}$ is ...

3
votes

1
answer

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### Numbers $n$ whose representation as the product of two divisors require more digits than that of $n$

Note: Posting in MO since it was unanswered in MSE
Let $f(x)$ be the number of digits in the decimal representation of $x$ e.g. $, f(0) = 1, f(1729) = 4$. If $n = ab$ then we can show that $f(ab) > ...

5
votes

0
answers

525
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### A generalization of the difference of two squares identity

This question deals with finding explicit integer functions for the coefficients of the monomial expansion of the polynomial
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\kappa_1, \ldots , \kappa_{n-...

6
votes

1
answer

205
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### Sequence A76132 eventually periodic modulo $2,3$ and $5$

Sequence A76132 starting as $1,1,2,4,10,36,218,\ldots$ of the OEIS is recursively defined by $a(1)=1$
and $a(n)=\sum_{k=1}^{n-1}a(n-k)^k$ for $n\geq 2$.
It is eventually periodic of period 1,1 and 34 ...

5
votes

1
answer

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### A common combinatorial description for a certain type of recurrences

For integer-valued sequences $(x_n)_{n=0}^\infty$, consider recurrences of the form
$$x_n=ax_{n-1}+(bn+c)x_{n-2} \tag{$*$}\label{star}$$
for $n\ge2$, where $a,b,c$ are integers.
There seem to be many ...

3
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0
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192
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### Is this sequence always periodical?

Is the following sequence always periodical?

2
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0
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### How to compute/estimate the least $k$ such that there exist $n$ consecutive integers each having a prime factor $\le k$?

Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$...

1
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1
answer

407
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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 ...

6
votes

0
answers

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### Is Collatz conjecture known to be true for specific 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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votes

1
answer

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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
votes

2
answers

138
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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 ...

21
votes

1
answer

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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 ...

4
votes

0
answers

323
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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 ...

4
votes

1
answer

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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;...

2
votes

1
answer

119
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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 ...

4
votes

1
answer

139
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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 ...

3
votes

1
answer

256
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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 $n$ ...

0
votes

1
answer

181
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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 ...

8
votes

1
answer

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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 ...

0
votes

1
answer

418
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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 ...

6
votes

5
answers

494
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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 $\exp(o(n))$?
...

1
vote

1
answer

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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 $(...

0
votes

0
answers

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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 ...

14
votes

4
answers

2k
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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 $a_n$ ...

5
votes

1
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248
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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 ...

6
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0
answers

277
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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) = ...

3
votes

1
answer

84
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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 ...

2
votes

1
answer

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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
...

5
votes

1
answer

203
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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).$...

3
votes

1
answer

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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^...

5
votes

1
answer

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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 ...

1
vote

1
answer

217
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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}=...

2
votes

2
answers

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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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0
answers

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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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0
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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 $\...

2
votes

0
answers

117
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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 \...

12
votes

1
answer

365
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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 ...

5
votes

1
answer

174
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### 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. ...

3
votes

1
answer

271
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 ...

32
votes

0
answers

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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 ...

17
votes

2
answers

908
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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:
$a(n)$ ...