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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Consecutive numbers with mutually distinct exponents in their canonical prime factorization
Is it possible to find 23 consecutive positive integers each of which has mutually distinct exponents in its canonical prime factorization? Such numbers are sequence A130091 in OEIS. 24 such numbers ...
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4th Order Floretions: Floret's Equation [closed]
Update: I've marked this question as answered. If you are thinking "What the heck are floretions?", go right to the answer provided by the Grinch. I definitely should have added clearer information on ...
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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 ...
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Making integer multisets graphic
Let $M=(X,f)$ be a multiset, where $X$ is the underlying set of elements and $f:X\rightarrow\mathbb{N}$ is the multiplicity function. For every $k\in\mathbb{N}$ put $k\cdot M:=(X,k\cdot f)$. It is ...
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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 ...
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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
$$
A_N(v) = \# \Big\{ (k,\ell) \in \{1, \dots, N\}^2, ~k \neq \ell:\quad n_k - n_\ell = v \Big\}.
$...
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Cardinal of a set cinsist of product of two sets?
Let
$$
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,\ ...
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A Collatz-like question about permutations
An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting.
Consider all permutations $\pi$ on the natural numbers such that ...
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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 $2$.
...
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Conway's subprime Fibonacci sequences
I want to be certain I have the latest information on
Conway's subprime Fibonacci sequences,
arXiv-posted a year ago; I am referencing the status in
a review.
To wit, starting with $(0,1)$:1
$$
0, 1, ...
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Searching information on a certain function with a fixed point property connecting Moebius $\mu$ and Fibonacci numbers
Let $\mu$ be the Moebius function and define for $1\leq n\in\mathbb{N}$
$$
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, \\
...
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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 $R(N,L)$ ...
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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(...
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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 $\{...
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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 ...
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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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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 ...
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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 ...
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Mod sequences that seem to become constant; and the number 316
Define a "mod sequence" of nonnegative integers
based on one start parameter $s$, its first term,
as follows.
$A(s)=(a_1,a_2,\ldots,a_n,\ldots)$
with $a_1 = s$
and
$$ a_n = \left(\sum_{k=1}^{n-1} a_k \...
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Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?
$F_n$ are the Fibonacci numbers.
In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$
satisfies:
$$ 4 \Phi_n(x)=A_n(x)^2 - (-1)^{(n-...
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1
answer
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Tower-of-squares sequence divides linear recurrent A001921 sequence?
Let $(a_n)$ be the A001921 sequence
$$
a_0 = 0,\ a_1 = 7, \quad a_{n+2} = 14a_{n+1} - a_n + 6.
$$
Let $(b_k)$ be the (almost)"tower-of-squares" sequence defined by
$$
b_0=2, \quad b_{k+1}=2b_k^...
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When is the ratio of Jacobi theta functions algebraic?
Probably this is well known.
$\theta_2$ and $\theta_3$ are Jacobi theta functions
as defined in mathworld (31) and (32).
For natural $n$ define
$$ f(n) = \frac{\theta_2(-e^{-\pi\sqrt{n}})}{\theta_3(-e^...
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votes
1
answer
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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 ...
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answers
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Strange (or stupid) arithmetic derivation
Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
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Arbitrarily large $n$ divides $F_n$
Is it true that there exists $n \in \mathbb{N}$ with arbitrarily many prime factors such that $n$ divides $F_n$, where $F_n$ represents the n-th Fibonacci number?
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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:
...
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1
answer
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views
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"...
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answers
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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 ...
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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, ...
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Are the asymptotics of A003238 known?
Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins
1, 1, 2, 3, 5, 6, 10, 11, 16, ...
and it is ...
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vote
1
answer
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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 ...
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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} \...
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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 ...
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votes
1
answer
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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 ...
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answers
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Sequences with integral means
Let $S(n)$ be the sequence whose first element is $n$, and from then onward,
the next element is the smallest natural number ${\ge}1$ that ensures that the
mean of all the numbers in the sequence is ...
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Is the sequence $a_n=c a_{n-1} - a_{n-2}$ always composite for $n > 5$?
Numerical evidence suggests the following.
For $c \in \mathbb{N}, c > 2$ define the sequence $a_n$ by
$a_0=0,a_1=1, \; a_n=c a_{n-1} - a_{n-2}$
For $ 5 < n < 500, \; 2 < c < 100$ there ...
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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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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 ...
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votes
3
answers
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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$ ...
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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\}$$
which is not in the OEIS. ...
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sequence, such that sum of any combinations in the sequence does not equal another [closed]
Hi,
Is there any known sequence such that the sum of a combination of one subsequence never equals another subsequence sum. The subsequences should have elements only from the parent sequence.
...
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Number of Configurations in the optimal Hanoi tower
There is a unique strategy how to move $n$ disks from the first rod to the second optimally and it takes $2^n-1$ steps, solution is obtained by simple recursion. I am interested into the following ...
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Maximal difference between k randomly drawn numbers from 1 to n – Looking for formula to sequence
Hello!
I have an interesting problem that seemed simple to me, but I'm unable to solve it on my own.
Suppose I am drawing k numbers out of n numbers labeled from 1 to n.
Considering all $\binom{n}{k}$...
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Which $n$ maximize $G(n)=\frac{\sigma(n)}{n \log \log n}$?
By Robin's theorem
$$G(n)=\frac{\sigma(n)}{n \log \log n}$$
is bounded by $e^\gamma \approx 1.78107241799$ for $n>5040$ assuming Riemann hypothesis .
For $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ ...
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votes
2
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An interesting, simple, sequence - surprised to find little material. [closed]
I've been considering this sequence:
$$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, ...
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votes
2
answers
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views
Maximal number of edges and triangular cells for n points in a triangular lattice
Consider a subset of $n$ points in an equilateral triangular lattice. Draw all the edges between nearest-neighbor points.
What is the maximum, over all such subsets, of the number of edges? This ...
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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 ...
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votes
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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) \...
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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 ...
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