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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On doubling or addition formulas for the sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$
We are interested which integer sequences are efficiently computable
possibly over finite rings.
Define the integer sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$
with initial terms $a(0),a(1)$...
- 25.4k
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157
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Conjecture: $x^4+1$ is never Wieferich prime
Related to this question and Alexander Kalmynin's answer.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, just to
avoid triviality ...
- 25.4k
2
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0
answers
110
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bijection from vectors with non-negative integer integer entries to integers
I have the following question. Given a natural number $N$ we construct a set $K$ of vectors of infinite length with non-negative integer entries with a given sum $N$. For example, for $N=3$ the set $K$...
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Interesting conjecture by Sequence Machine
Let $a(n)$ be A344960 (i.e., position of binary complement of $n$-th word in A341258). By definition, in order to calculate $a(n)$, we need to know A341258. Below we will correspond this sequence with ...
- 4,948
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32
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joint rank sequences
An algebraic question I have been working on led me to a sequence that appears in OEIS as A186355: "adjusted joint rank sequence of $(f(i))$ and $(g(j))$ with $f(i)$ before $g(j)$ when $f(i)=g(j)$...
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72
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Possible subsequence of the A110978
Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
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199
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Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
- 4,948
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0
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126
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Recurrence for A004208
Let $a(n)$ be A004208. Here
$$a(n)=n\prod\limits_{j=1}^{n}(2j-1)-\sum\limits_{i=1}^{n-1}a(i)\prod\limits_{j=1}^{n-i}(2j-1)$$
I conjecture that
$$a(n)=R(n-1,0)$$
where
$$R(n,q)=2(q+2)R(n-1,q+1)+\sum\...
- 4,948
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100
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Sequence of numbers related to line-segment intersections
Question:
what is known about the sequence $\mathbb{X}\subset \mathbb{N}_0$ such that for each $k\in \mathbb{X}$ there exists a set of $n$ points in general position in the Euclidean plane such that ...
- 13.2k
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0
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105
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Sequences that sum up to the many sequences in the OEIS
Let
$$a(n,m,k)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj\geqslant 0]\binom{n}{j}\binom{n+kj}{j-1}(m-1)^{j-1}$$
Here square brackets denote Iverson brackets.
There are many sequences in the OEIS that are ...
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0
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70
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Integer coefficients such that $T(n,k)=R(n,k)-R(n,k-1)$
Let $a(n)$ be A000085, i.e., the number of self-inverse permutations on $n$ letters, also known as involutions; number of standard Young tableaux with $n$ cells. Here
$$a(n) = a(n-1) + (n-1)a(n-2), a(...
- 4,948
2
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0
answers
76
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Uniqueness of the permutation
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
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0
answers
157
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Closed form for the A347205
Let $q(n)$ be A007814, i.e., number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $\operatorname{wt}(n)$ be A000120, i.e., number of $1$'s in binary ...
- 4,948
2
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0
answers
115
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Closed form for the sum of the integer coefficients
Let $a(n)$ be A002720, i.e., number of partial permutations of an $n$-set; number of $n \times n$ binary matrices with at most one $1$ in each row and column.
$$a(n)=\sum\limits_{k=0}^{n} k!\binom{n}{...
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2
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239
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Chess pieces metrics in higher dimensions
A couple of days ago, I was thinking about applying the knight (the well-known piece of chess) metric to any cubic lattice $\mathbb{N}^k$, $k \in \mathbb{N}-\{0,1\}$.
I suddenly realized that, from $k ...
- 1,451
2
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1
answer
174
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Asymptotic analysis of a peculiar sum of squares sequence
Let $a,b$ be two positive integers. Let the sequence $\{s_n\}_n$ be the set of all possible sums of squares $a^2+b^2$, such that they are in ascending order
\begin{align*}
& n=1 & s_1=1^2+1^2=...
- 151
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108
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How to compute/estimate the least $k$ such that there exist $n$ consecutive integers each having a prime factor $\le k$?
Let $a_n$ be the least integer $k$ such that there exist $n$ consecutive integers each with a prime factor $\le k$. For example, $a_{13} \le 11$ because the 13 consecutive integers $114,115,\ldots,126$...
- 523
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137
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Writing integers as sequences of products by 2 and integer divisions by 3
For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$.
For instance:
$$
100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
- 1,444
2
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0
answers
327
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Why can one compute the sum of divisors of $n$ without factoring $n$?
Question links to paper
which states:
$$
\sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1)
$$
where $\sigma(n)$ is the sum of divisors of $n$.
Another similar ...
- 25.4k
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0
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176
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A question on $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$
Let $a_i(n) = a_i(\pi(n)) + a_i(n-\pi(n))$ with $a_i(n) = 1$ for $n \le i$ where $\pi(n)$ is the prime-counting function.
By definition, it is obvious that $a_1(n) = n$ and $a_2(n)$ is https://oeis....
- 701
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0
answers
98
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Two conjectures inspired from an equation involving the sum of divisors and the Euler's totient function due to Iannucci
In this post I add two equations involving the sum of divisors $\sigma(n)$ and the Euler's totient function, denoted in this post as $\varphi(n)$, and after I ask about a conjecture involving these. ...
2
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0
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120
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Sieving the values of an arithmetic sequence which is infinitely many times $1$
I have a sequence of positive integers $a_n$ which assumes infinitely many times the value $1$. I want to estimate the cardinality of the following set:
$$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
2
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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
2
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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
2
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0
answers
154
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Equi-distribution of the parity of partitions
The integer partition function $p(n)$ has a generating function given by
$$\frac1{(q)_{\infty}}=\sum_{n=0}^{\infty}p(n)q^n$$
with $(q)_{\infty}=\prod_{m=1}^{\infty}(1-q^m)$. The long-standing problem ...
- 43.2k
2
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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
2
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0
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75
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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
2
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0
answers
311
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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.
$$n=\prod_{i=1}^{k}p_i^{\alpha_i} \...
- 442
1
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1
answer
123
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Given a real $x>1$, construct an aperiodic substitution sequence whose complexity functions grow like $xn$
The Fibonacci word is a binary sequence defined as follows.
We use a substitution rule $0\to 01$, $1\to 0$. Then, starting with the binary string $0$, apply the substitution rules successively. So we ...
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1
vote
1
answer
115
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Cardinality of $\{ n_i + i^k: i \in \mathbb{N} \} \cap [1,T]$ where $\{n_i \}$ is all natural numbers in some order
Let $n_1, n_2, ...$ be a sequence of natural numbers such that $\{n_i: i \in \mathbb{N}\}$ as a set is all of natural numbers. Let $k$ be a positive integer. Is is possible to obtain a lower bound of ...
- 3,625
1
vote
1
answer
181
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On the sequence $a(n)=\gcd(2^n-1,\phi(2^n-1))$
For natural $n$, define the sequence
$$
a(n)=\gcd(2^n-1,\phi(2^n-1))
$$
It doesn't appear to be in OEIS and starts
$1,1,1,1,9,1,1,1,3,1,9,1,3,1,1,1,27,1,75,49$
Q1 Can we unconditionally prove $a(n)=1$...
- 25.4k
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1
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Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$?
I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers.
Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^...
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2
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183
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Weirdness in the sequence "the number of divisors for a weird number"
I thought it would be fun to give my froshling students a short programming assignment to characterize numbers as: deficient, abundant, perfect, and prime. Then I got a little carried away and started ...
- 11
1
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1
answer
176
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The sequence $G(n,k)=G(n-2,k)+G(n,k-2)$
Background: The binomial coefficients $C(n,k)$ satisfy the recurrence
$C(n,k)=C(n-1,k)+C(n-1,k-1)$ and some terminating conditions, for
more information check here.
$C(n,k)$ doesn't appear to be ...
- 25.4k
1
vote
2
answers
534
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Can these sequences stay integer-valued as many times as we want and then fail?
Edit:
Suppose that we choose some integer $d$ and some natural number $c=c_2$. Then if we plug those values into
$$ c_{n+1}=\frac{c_n(c_n+n+d)}n $$ and observe the behavior of this recursively ...
user114642
1
vote
1
answer
221
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Correctness of the algorithm for the A329369, A347205 and related sequences
Let $a(n)$ be A347205. It is enough for us to know that
$$
a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^jk), \\
a(0) = 1
$$
Let $b(n)$ be A329369. It is enough for us to know that
$$
b(2^m(2k+1)) = \sum\...
- 4,948
1
vote
2
answers
218
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Values of the Kronecker symbol of recursive sequences
Is there something known about the values of the Kronecker symbol $\left(\frac{a_n}{a_{n+1}}\right)$ if $a_n$ is a recursive sequence? I know something about this if $a_n$ is the Fibonacci sequence or ...
- 1,101
1
vote
1
answer
147
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Strongly regular binary sequences
Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote the set of non-negative integers. If $n\in\mathbb{N}$ we let $[n] = \{0,\ldots,n-1\}$. For $A
\subseteq \mathbb{N}$ we let $$\mu^+(A) = \lim\sup_{n\to\infty}\...
- 48.9k
1
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2
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390
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Are there infinitely long arithmetic progressions in every increasing sequence of positive integers with bounded gaps between consecutive terms?
Suppose the largest gap is D>1 and at least two of the gaps 1,2,...,D appear infinitely many times. I think the answer is NO. But I find it difficult to formulate a necessary and sufficient ...
- 53
1
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1
answer
1k
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How many non-isomorphic, simple, connected graphs with 6 vertices are there? [closed]
A graph is called simple if there are no loops and there are no multiple edges. Is it possible to compute the number of non-isomorphic, simple, connected graphs with 6 vertices? If the number is known,...
- 331
1
vote
1
answer
194
views
Does the Kimberling sequence map numbers "arbitrarily far away"?
The Kimberling sequence is a recursively defined "shuffling sequence" (pictorial description here). Let $k:\mathbb{N}\to \mathbb{N}$ be the Kimberling sequence. Does $k$ map members of $\mathbb{N}$ ...
- 48.9k
1
vote
1
answer
276
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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 ...
- 25.4k
1
vote
1
answer
77
views
Sequence derived from transform of a given vector (with Fibonacci as partial sums)
Let F_n be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1
$$
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). ...
- 4,948
1
vote
1
answer
114
views
Coefficients of number of the same terms which are arising from iterations based on binary expansion of $n$
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
Here $T(n,k)$ is the $(k+1)$-th bit from the right side in the binary ...
- 4,948
1
vote
1
answer
594
views
Polynomials, $3^x$ and the Collatz conjecture
$\DeclareMathOperator\Orb{Orb}\newcommand\abs[1]{\lvert#1\rvert}$The Collatz or the $3n+1$ conjecture is open.
Are there non-trivial polynomials $f(x)\in\mathbb Z[x]$ and $g(x)\in\mathbb R[x]$ having ...
- 13.9k
1
vote
1
answer
280
views
Sequences over finite fields
Let's we have finite field $F_q$ for some prime $q=2^M-1$.
I am looking for special sequence {$a_{i}$, $i \in {1,..,q-1}$},
($\{a_{1},...,a_{q-1}\}=F_q/\{0\}$) with the following properties:
$r_{1}=...
- 33
1
vote
1
answer
229
views
constructing a covering system of congruences?
A family of residue classes $a_i (\mod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$ is called a covering system of congruences if every integer belongs to at least one of the residue classes, that is, ...
- 841
1
vote
1
answer
163
views
How many points appear in the plane when the chain of n-gons is close?
Let $A_{11}A_{12}\cdots A_{1n}$ be a regular $n$ polygon, we call $A_{11}A_{12}\cdots A_{1n}$ is the $1st-n-gons$. Now we construct the $2nd-n-gon$ based two condition as follows:
$2nd-n-gons$ is ...
- 1,776
1
vote
1
answer
226
views
An elementary sequence question [closed]
Below is a problem, from an old Silk Road olympiad.
Define an infinite sequence, $a(n)$, such that, $a(1)=a(2)=1$;
$$
a(n)=a(a(n-1))+a(n-a(n-1)),\forall n\geq 3.
$$
Show that, for every $n\geq 1$, $a(...
- 59
1
vote
1
answer
125
views
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}^...
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