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

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0
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1answer
216 views

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 ...
1
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0answers
81 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 ...
2
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3answers
206 views

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 ...
4
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0answers
144 views

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 ...
2
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0answers
111 views

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\}. ...
3
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0answers
56 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,\ ...
5
votes
1answer
479 views

Intuition behind salient numbers in number of h-cobordism classes of smooth homotopy n-spheres

The Wikipedia article on Exotic Sphere displays this sequence of numbers (see also OEIS A001676 and the Milnor link therein) for the order of the classses as $$1, \;1, \;1,\; 1,\; 1, \;1, \;28,\; ...
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3answers
261 views

Upper bound on length of addition chain

An addition chain for $n$ is a finite sequence of integers starting at 1 and ending at $n$, such that each element is a sum of two previous elements. A short addition chain for $n$ can be used, for ...
7
votes
1answer
309 views

The p-adic valuation of a linear recurrence

Let $(u_n)_{n \geq 0}$ be an integer-valued linear recurrence of order $k \geq 1$. Precisely, $$u_n = a_1 u_{n-1} + \cdots + a_k u_{n - k} \quad \forall n \geq k ,$$ for some $a_1, \ldots, a_k \in ...
1
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2answers
130 views

Values of the Kronecker symbol of recursvie sequences

Is there something known about the values of the Kronecker symbol $\left(\frac{a_n}{a_{n+1}}\right)$ if $a_n$ is a recursive sequence? I know something about this if $a_n$ is the Fibonacci sequence or ...
6
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1answer
279 views

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, ...
11
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1answer
211 views

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 ...
0
votes
1answer
93 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 $2$. ...
3
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0answers
50 views

Increasing integral sequence of intermediate growth which is periodic modulo almost all primes

Many integral sequences are periodic modulo (almost) all primes. However all examples I know are either evaluations of suitable polynomials on consecutive integers (trivial examples) or grow at least ...
4
votes
4answers
461 views

What is the motivation and purpose of the Floretion group?

When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
3
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0answers
78 views

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, \\ ...
0
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0answers
77 views

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 $R(N,L)$ ...
5
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1answer
158 views

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 ...
2
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0answers
149 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 ...
9
votes
2answers
729 views

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 ...
0
votes
1answer
167 views

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) = ...
12
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0answers
326 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 ...
3
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0answers
230 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 ...
35
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1answer
930 views

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 ...
4
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1answer
177 views

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 - ...
2
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1answer
153 views

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 ...
6
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0answers
133 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) = ...
5
votes
1answer
491 views

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?
-2
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1answer
150 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 ...
2
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0answers
57 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
vote
1answer
187 views

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 ...
29
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1answer
1k views

Can we find lattice polyhedra with faces of area 1,2,3,…?

I asked this question two months ago on MSE, where it earned the rare Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days. Perhaps justifiably so! Here I repeat it with ...
7
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1answer
405 views

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 ...
13
votes
1answer
526 views

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 ...
5
votes
1answer
358 views

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: ...
1
vote
1answer
226 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 ...
2
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0answers
288 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} ...
2
votes
1answer
114 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 ...
8
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1answer
398 views

Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$

I am currently interested in the following sequence: $$\begin{cases}u_0 & = & \alpha\\u_n & = & u_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275... $ with $C$ being the ...
3
votes
1answer
275 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 ...
21
votes
5answers
911 views

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 ...
13
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6answers
807 views

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$ ...
48
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4answers
4k views

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 ...
0
votes
1answer
186 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 ...
6
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0answers
389 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 ...
3
votes
2answers
152 views

Databases for sequences indexed by partitions

Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such ...
1
vote
1answer
174 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. ...
5
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0answers
379 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 ...
0
votes
2answers
199 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. ...
11
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0answers
592 views

Is OEIS A007018 really a subsequence of squarefree numbers?

A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004 Is it really so? As far as I know, it is open problem ...