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

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3
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0answers
24 views

Current 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 ...
6
votes
0answers
175 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
vote
1answer
83 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
votes
1answer
234 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, ...
10
votes
1answer
180 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
86 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
votes
0answers
45 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
3answers
309 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
64 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
votes
0answers
68 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
votes
1answer
146 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
146 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 ...
0
votes
1answer
159 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
317 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
votes
0answers
225 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 ...
34
votes
1answer
914 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
votes
1answer
171 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
votes
1answer
151 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
votes
0answers
130 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
426 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
votes
0answers
54 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
182 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 ...
28
votes
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
votes
1answer
400 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
507 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
352 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
212 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
votes
0answers
283 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
111 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
votes
1answer
387 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
871 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
votes
6answers
765 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
votes
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
173 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
votes
0answers
357 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
147 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
168 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
votes
0answers
365 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
196 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
votes
0answers
573 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 ...
8
votes
2answers
999 views

sequences with a fractal dimension

This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...
2
votes
3answers
533 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
vote
1answer
296 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 ...
3
votes
3answers
877 views

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
2answers
666 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, ...
7
votes
2answers
569 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 ...
18
votes
3answers
1k views

Zeroes of the random Fibonacci sequence

Let X_n be the "random Fibonacci sequence," defined as follows: $X_0 = 0, X_1 = 1$; $X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips. It is known that ...
14
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1answer
1k views

Some unpublished notes of Hofstadter

I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because ...