Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
2
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Coefficients for Powers of the Mittag-Leffler Function
Considering the one parameter Mittag-Leffler function,
$$E_{\alpha}(z)=\sum_{k=0}^\infty\frac{z^{k}}{\Gamma(\alpha k+1)}, \Re(\alpha)>0$$
Considering then the generating function for $E_\alpha(z^...
2
votes
1
answer
154
views
GCD for two Cullen numbers
The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
- 21
0
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0
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315
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Number Theory and p-Power-Partitioned Numbers
Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, we're going to define its digits-partition as the next set $D_{N} = \bigcup_{j=1}^{n}\bigcup_{k=1}^{p(a_{j})}\{(P_{k},j)\}$, where each pair $(...
user89024
4
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answer
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Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?
$F_n$ are the Fibonacci numbers.
In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$
satisfies:
$$ 4 \Phi_n(x)=A_n(x)^2 - (-1)^{(n-...
- 25.4k
1
vote
1
answer
81
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Reference request for multiple free sequences
Erdos usually named a sequence of integers no one of which is divisible by any other as an $M$- sequence (M stands for "multiple-free") or primitive sequence.
For example it is easy to see that $\...
- 3,876
5
votes
5
answers
677
views
Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$?
Can an integer or rational sequence satisfy some bounded order recurrence $\mod \ $ almost all primes but doesn't satisfy such in $\mathbb{Q}$?
The recurrences $\mod p$ can be different, possibly ...
- 25.4k
1
vote
1
answer
125
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The connection between the length of Fibonacci $p$-step numbers and its limit values
One of the most important generalization of the classical Fibonacci numbers is the Fibonacci $p$-step numbers that is defined as follows
\begin{equation}\label{cp26}
F_n^{(p)}=F_{n-1}^{(p)}+F_{n-2}^...
- 313
1
vote
0
answers
116
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In search of multiple expressions for a sequence
The sequence $a_n=\sum_{k=0}^n\binom{n}k^24^k$ is listed on OEIS along with a couple of combinatorial interpretations. What interested me at the moment is the plethora of binomial single-sums for the ...
- 43.2k
7
votes
1
answer
283
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On one class of Somos-like sequences
This question is motivated by integrability of the sequence mistakenly arisen in the question Does this sequence always give an integer?
Let $m_1,\ldots, m_{k-1}$ be positive integers and sequence $\{...
- 12.3k
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2
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Which $n$ maximize $G(n)=\frac{\sigma(n)}{n \log \log n}$?
By Robin's theorem
$$G(n)=\frac{\sigma(n)}{n \log \log n}$$
is bounded by $e^\gamma \approx 1.78107241799$ for $n>5040$ assuming Riemann hypothesis .
For $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ ...
- 25.4k
3
votes
1
answer
91
views
Source coding lexicographic index of finite alphabet sequence with weight (partitions)
My goal is to determine the lexicographic index of an $M$-ary $n$-sequence $\mathbf{x}$ on the subset with an $M$-weight sum constraint:
$$S = \{ \mathbf{x} \in \{0, \ldots, M-1\}^n: \sum_{j=1}^n x_j =...
- 131
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0
answers
385
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A strange polynomial equality
In my answer to this question, I have obtained that the polynomial $p(x)$ of degree $2n$ with nonnegative values on $[-1,1]$ with $p(\pm1)=1$ has $\int_{-1}^1 p(x)\,dx\geq \frac{4}{(n+1)(n+2)}$, and ...
- 23.7k
2
votes
0
answers
55
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Collinearity in Enumerations of the Rationals
I am looking for a solution of the No Three-in-a-Line problem for the whole $\mathbb{Z}\times\mathbb{Z}$ plane and had the idea, to use a non-redundant enumeration of the rationals, like the breadth-...
- 13.2k
1
vote
0
answers
390
views
Is there a "complete" Sidon sequence?
A sequence of natural numbers $(a_n)$ with the property that all pairwise sums of elements are distinct is called a Sidon sequence and it is proved there are at most $s(n)\sim\sqrt n$ elements of ...
- 3,876
7
votes
1
answer
455
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More asymptotics for trees
This is a follow up to my recent question on the asymptotics of A003238. Lucia gave a fine answer to that question, but as I hinted the 'real' problem I have in mind is slightly different, and I've ...
- 1,305
15
votes
1
answer
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Conjecture on signed sum of integer fractions x/y from 1..N?
Here is a generalization of an integer challenge that was asked on Yahoo!Answers in 2009, I believe it could be original, defies induction and has exponential-complexity. Not aware of any theory that ...
3
votes
3
answers
696
views
For any prime $p$, is there $C$ such that if $x\ge C$, then all but one integer among $x+1, x+2, \dots, x+p$ has Greatest Prime Factor $> p$
I apologize if this is a naive question about greatest prime factors (gpf). I was thinking about the sequence of integers where $\mathrm{gpf}(x) \le p$ where $p$ is any prime.
Clearly, as $x$ ...
- 1,049
2
votes
0
answers
70
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Do almost all zeros of linear recurrence come from scaling or cancellation?
Let $a(n)$ be linear recurrence with constant coefficient of order $t$.
Assume $a(n)=\sum_{i=0}^t c_i r_i^n$ where $r_i$ are the roots of
the companion polynomial and $c_i$ are algebraic numbers.
...
- 25.4k
15
votes
0
answers
487
views
Word complexity of primes mod 4
For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
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0
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0
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93
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What is the class of real sequences satisfying these conditions?
I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions:
$\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\...
7
votes
1
answer
125
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Integer Recursion Reference Request
I've run across the following recursion which at times seems very steady and predictable and at other times seems very chaotic.
Let $c_1, \dots c_k, b_0, m \in \mathbb{Z}$ with $b_0>m\ge 3$ and $...
- 2,235
0
votes
0
answers
81
views
Terminology for "approximately convex" sequence of integers
I have a sequence of integers meeting the following inequality:
$u_n \leq \frac{u_{n-1}+u_{n+1}}{2} + \frac{1}{2}$. In other words, the sequence is "approximately convex", and the difference comes ...
- 101
0
votes
1
answer
219
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A square-squareroot integer race sequence involving primes
I wonder what is the expected behavior of this process?
Let
$f^2_{\mathrm{next}}(n) =$ the next prime after $n^2$.
$g_{\mathrm{sqrt}}(n) = \lfloor \sqrt{n} \rfloor$.
Now iterate as follows, ...
- 151k
0
votes
1
answer
134
views
How to write a given rank matrix with some constraints?
If I want to write an $m\times n$ $0/1$ matrix with only rows or columns distinct, I could just pick $m$ or $n$ distinct natural numbers effectively writing them down as rows or columns in base $2$.
...
- 13.9k
5
votes
1
answer
384
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Flow of an integer
I've stumbled across this family of flow networks, and posted the sequence of maximal flows to OEIS: A238729. I can't find any reference to it either. Has anyone seen it?
Here is the description:
...
3
votes
0
answers
180
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Additive combinatorics and a Diophantine equation
Let $(n_k)_{1 \leq k \leq N}$ be a sequence of distinct positive integers. For $v \in \mathbb{Z}$ set
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
$...
- 2,227
0
votes
1
answer
215
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Number of squares in a grid under certain conditions
Consider an $(n+1)\times (n+1)$ grid of lattice points in the plane.
$A(n):$ # of squares with vertices on the grid.
It's relatively well-known that $A(n)=\frac{n(n+1)^2(n+2)}{12}$. Now, $A(n) = B(...
0
votes
0
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224
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Classic question on integer partitions (with distinct summands)
I guess that the following was solved sometime in the 18th century, but could not find a reference to it. I am interested in approximations to the following integer partition problem:
Denote $R(N,L)$ ...
- 1
1
vote
1
answer
276
views
Infinitely many sufficiently large powers in linear recurrences
Edit Aaron solved the original question with the
fourth order $$ a(n)=n2^n+\frac{(-1)^n-1^n}{2} $$
trying to make the question harder.
Let $a(n)$ be a linear recurrence with constant coefficients,
of ...
- 25.4k
2
votes
0
answers
213
views
Avoiding Fibonacci-like sequences
Suppose we are trying to avoid 3-term arithmetic progressions. There are two relevant sequences in the OEIS pertaining to this:
A003278: The sequence whose $n^{\text{th}}$ term is the smallest number ...
- 4,462
3
votes
0
answers
252
views
What are the values of this sequence?
Let $F_n$ denote the $n$th Fibonacci number.
Then $\prod\limits_{i=1}^{\infty}(1-x^{F_i})$ is a series all of whose coefficients are either $-1$, $0$ or $+1$.
The sequence of the coefficients in ...
- 2,143
3
votes
1
answer
298
views
Sequences with integral variances
This is a companion to my earlier question,
Sequences with integral means.
This new question is, frankly, not as interesting, but it feels necessary to complete
the thought.
Let $V(n)$ be the ...
- 151k
2
votes
1
answer
252
views
Is there a linear recurrence with infinitely many zeros, conjecturally infinitely many primes and non-zero terms of exponential growth?
Let $a_n$ be a linear recurrence with integer constant coefficients
and initial values.
Is it possible $a_n$ to satisfy all of these:
$a_n = 0$ infinitely often.
if $a_n \ne 0$, $ | a_n |$ is of ...
- 25.4k
7
votes
0
answers
557
views
Graphs with graphic imbalance sequences
Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
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3
votes
0
answers
102
views
Cardinal of a set cinsist of product of two sets?
Let
$$
A=\{1,2,\ldots,p-1\},\qquad B=\{1,2,\ldots,q-1\}
$$
where $p,q$ are primes not necessarily distinct.
Is there any elementary way to find the cardinal of the following set
$$
AB=\{ab:\ a\in A,\ ...
- 841
6
votes
0
answers
207
views
When is the ratio of Jacobi theta functions algebraic?
Probably this is well known.
$\theta_2$ and $\theta_3$ are Jacobi theta functions
as defined in mathworld (31) and (32).
For natural $n$ define
$$ f(n) = \frac{\theta_2(-e^{-\pi\sqrt{n}})}{\theta_3(-e^...
- 25.4k
0
votes
1
answer
250
views
What is a description of winning strategies in this tile game?
I'm hoping someone can help me figure out how to describe all winning strategies for "Player 1" in the following game:
Consider a board with $n$ tiles arranged in a row. Player 1 and Player 2 each ...
- 11
6
votes
0
answers
669
views
Number of Configurations in the optimal Hanoi tower
There is a unique strategy how to move $n$ disks from the first rod to the second optimally and it takes $2^n-1$ steps, solution is obtained by simple recursion. I am interested into the following ...
- 399
-2
votes
1
answer
180
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Decimal digits multiplied by powers of 2: leads to mod 8? [closed]
This is more a puzzle than a research question,
a puzzle to me. Perhaps it is straightforward for others.
Imagine Repeatedly interpreting a number
expressed with the usual base-$10$ digits
as "digits"...
- 151k
2
votes
0
answers
311
views
A question concerning the strange arithmetic derivation
This question is related to Strange (or stupid) arithmetic derivation. The original question whether an unbounded sequence of iterates exists is still unanswered.
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \...
- 442
3
votes
0
answers
100
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Searching information on a certain function with a fixed point property connecting Moebius $\mu$ and Fibonacci numbers
Let $\mu$ be the Moebius function and define for $1\leq n\in\mathbb{N}$
$$
f(n) =
\left\{
\begin{array}{ll}
\mu\left(\frac{n}{2}\right) + \mu\left(\frac{n}{4}\right), & n\equiv 0, 4, 8\mod 12, \\
...
- 410
1
vote
1
answer
236
views
Nested Sequence of Integers
In some combinatorial research I came across the following nested sequence:
$$\{a_n\}=\{1,1,3,1,7,3,17,1,35,7,77,3,157,17,331,1,663,35,1361,7,2729,77,5535,3,11073, \dots\}$$
which is not in the OEIS. ...
- 2,235
0
votes
2
answers
226
views
sequence, such that sum of any combinations in the sequence does not equal another [closed]
Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.
...
- 119
1
vote
1
answer
334
views
Maximal difference between k randomly drawn numbers from 1 to n – Looking for formula to sequence
Hello!
I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own.
Suppose I am drawing k numbers out of n numbers labeled from 1 to n.
Considering all $\binom{n}{k}$...
- 13
-4
votes
2
answers
763
views
An interesting, simple, sequence - surprised to find little material. [closed]
I've been considering this sequence:
$$1,2,3,6,12,24,48,96,192,...$$
I've generated the sequence from the rule
$$V_n=\sum_{0\leq i \lt n} V_i$$
$$V_0=1; V_1=2V_0=V_0+V_0$$
What interests me most, ...
- 5
5
votes
0
answers
753
views
Least Prime Factor in a sequence of 2n consecutive integers
I was thinking about consecutive integers and I wondered if anyone had done work exploring whether a sequence of $2n$ consecutive integers (i.e. 101,102,103,...,100+2n) always contains at least one ...
- 1,049
13
votes
0
answers
718
views
Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable $\mod 4$?
Is A001935 Number of partitions with no even part repeated efficiently computable $\mod 4$?
I am interested because of this relation with sum of divisors of $8n+1$.
$\sigma(8n+1) \equiv A001935(n) \...
- 25.4k
2
votes
0
answers
75
views
Regular graphs with unimodal subdegrees that are not distance-regular
Distance regular graphs are known to exhibit the following property: starting from an arbitrary vertex $\alpha$, let $k_i$ denote the number of vertices at distance $i$ from $\alpha$ (in terms of ...
- 1,164
0
votes
0
answers
55
views
Sequences that sum up to sums of integer coefficients
Let
$$
T(n,k,p,q,r,s) = (q(k-1)+1)T(n-1,k,p,q,r,s) + s(n+r(k-1)+p-2)T(n-1,k-1,p,q,r,s), \\
T(n,1,p,q,r,s) = 1, \\
T(n,0,p,q,r,s) = T(0,k,p,q,r,s) = 0
$$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\...
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