Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
7
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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
2
votes
0
answers
213
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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 ...
- 4,462
10
votes
2
answers
1k
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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 ...
- 3,717
0
votes
1
answer
215
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) = B(...
15
votes
0
answers
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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 ...
- 17.1k
3
votes
0
answers
252
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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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41
votes
1
answer
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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
$$ a_n = \left(\sum_{k=1}^{n-1} a_k \...
- 151k
4
votes
1
answer
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Is $p$ is square modulo $F_p$ when $p=4k+1 > 5$?
$F_n$ are the Fibonacci numbers.
In On computing factors of cyclotomic polynomials p.1 for odd square-free $n>1$ the cyclotomic polynomial $\Phi_n(x)$
satisfies:
$$ 4 \Phi_n(x)=A_n(x)^2 - (-1)^{(n-...
- 25.4k
2
votes
1
answer
214
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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
$$
b_0=2, \quad b_{k+1}=2b_k^...
- 621
6
votes
0
answers
207
views
When is the ratio of Jacobi theta functions algebraic?
Probably this is well known.
$\theta_2$ and $\theta_3$ are Jacobi theta functions
as defined in mathworld (31) and (32).
For natural $n$ define
$$ f(n) = \frac{\theta_2(-e^{-\pi\sqrt{n}})}{\theta_3(-e^...
- 25.4k
5
votes
1
answer
737
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?
- 53
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1
answer
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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
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
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
41
votes
2
answers
2k
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 ...
- 151k
7
votes
1
answer
455
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 ...
- 1,305
14
votes
1
answer
697
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 ...
- 1,305
5
votes
1
answer
384
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
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
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.
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \...
- 442
2
votes
1
answer
252
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Is there a linear recurrence with infinitely many zeros, conjecturally infinitely many primes and non-zero terms of exponential growth?
Let $a_n$ be a linear recurrence with integer constant coefficients
and initial values.
Is it possible $a_n$ to satisfy all of these:
$a_n = 0$ infinitely often.
if $a_n \ne 0$, $ | a_n |$ is of ...
- 25.4k
12
votes
2
answers
1k
views
Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$
I am currently interested in the following sequence:
$$\begin{cases}u_0 & = & \alpha\\u_n & = & u_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275... $ with $C$ being the ...
- 123
3
votes
1
answer
298
views
Sequences with integral variances
This is a companion to my earlier question,
Sequences with integral means.
This new question is, frankly, not as interesting, but it feels necessary to complete
the thought.
Let $V(n)$ be the ...
- 151k
23
votes
5
answers
1k
views
Sequences with integral means
Let $S(n)$ be the sequence whose first element is $n$, and from then onward,
the next element is the smallest natural number ${\ge}1$ that ensures that the
mean of all the numbers in the sequence is ...
- 151k
14
votes
5
answers
977
views
Is the sequence $a_n=c a_{n-1} - a_{n-2}$ always composite for $n > 5$?
Numerical evidence suggests the following.
For $c \in \mathbb{N}, c > 2$ define the sequence $a_n$ by
$a_0=0,a_1=1, \; a_n=c a_{n-1} - a_{n-2}$
For $ 5 < n < 500, \; 2 < c < 100$ there ...
- 25.4k
49
votes
4
answers
4k
views
Strange (or stupid) arithmetic derivation
Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
- 3,004
0
votes
1
answer
250
views
What is a description of winning strategies in this tile game?
I'm hoping someone can help me figure out how to describe all winning strategies for "Player 1" in the following game:
Consider a board with $n$ tiles arranged in a row. Player 1 and Player 2 each ...
- 11
7
votes
0
answers
557
views
Graphs with graphic imbalance sequences
Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...
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4
votes
2
answers
240
views
Databases for sequences indexed by partitions
Is there a database for sequences indexed by partitions similar to Sloane's OEIS? I mean, I am aware that in the OEIS there are some arrays indexed by partitions, but I feel as though most of such ...
- 16.9k
1
vote
1
answer
236
views
Nested Sequence of Integers
In some combinatorial research I came across the following nested sequence:
$$\{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. ...
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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
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
19
votes
1
answer
1k
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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 that really so?
As far as I know, it is an open ...
- 25.4k
14
votes
2
answers
2k
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 ...
- 13.4k
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
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}$...
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4
votes
2
answers
2k
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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}$$
is bounded by $e^\gamma \approx 1.78107241799$ for $n>5040$ assuming Riemann hypothesis .
For $n=\mathrm {lcm} (1,2 \dots k)$, $G(n)$ ...
- 25.4k
-4
votes
2
answers
763
views
An interesting, simple, sequence - surprised to find little material. [closed]
I've been considering this sequence:
$$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
7
votes
2
answers
964
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 ...
- 567
23
votes
3
answers
2k
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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 ...
- 19.2k
17
votes
2
answers
3k
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 they'...
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
8
votes
4
answers
1k
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A Pascal's-triangle -like random process
I was exploring Pascal's triangle on a cylinder when I encountered this puzzle-like problem.
It is surely elementary, but perhaps weekend-entertaining.
Start with a permutation of $(1,2,3, \ldots, n)$...
- 151k
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
11
votes
1
answer
864
views
Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated )
Up to $10^6$:
$\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$
A001935 Number of partitions with no even part repeated
Is this true in general?
It would mean relation between restricted partitions ...
- 25.4k
14
votes
1
answer
4k
views
Put as many points as possible in an equilateral triangle of side 1 with their minimal distance greater than 1/n
It is known by the pigeon-hole principle that:
If we select $5$ points within an equilateral triangle with side $1$, there must be at least two whose distance apart is less than or equal to $1/2$.
...
- 241
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
15
votes
1
answer
2k
views
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 ...
20
votes
13
answers
7k
views
Longest coinciding pair of integer sequences known
There are arbitrarily many pairs of integer sequences (of arbitrary origins) that coincide upto an $N$ but differ for an $n > N$. I assume, the coincidence will be considered accidentally then by ...
- 9,720