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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Simply generated sequences with mysterious differences
Suppose that $a_0 < a_1,$ $b_0 < b_1,$ and $$a_n=a_1b_{n-1}+a_0b_{n-2}+qn+r$$ for $n \geq 2$, where $a_0,a_1,b_0,b_1,q,r$ are integers such that $(a_n)$ and $(b_n)$ are increasing and ${(|a_n|)}$...
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Elliptic curve sequences needed for universal forgery
Elliptic Curve Digital Signature Algorithm (ECDSA) admits universal forgery (UF) if the Attacker can solve the equation
$$z=\frac{f_{k-1}(x,y)f_{k+1}(x,y)}{f_{k}(x,y)^2},$$
where $k$ is unknown, $f_{k}...
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A possibly surprising appearance of $\sqrt{2}.$
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_1b_{n-1}-a_0b_{n-2} + 2n$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
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Limit associated with complementary sequences
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
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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(...
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Elementary description to count of perfect squares - II
What can we say about growth of smallest gap $g(a)$ which is the smallest $|x-y|$ where $0\leq x,y\leq\Big\lfloor\frac a2\Big\rfloor$ and $\sqrt{x(a-x)},\sqrt{y(a-y)}\in\Bbb Z$?
Is $g(a)=1\iff a=b^2+...
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Elementary description to count of perfect squares - I
Is there an elementary description of $$N(a)=\Big|\Big\{x\in\{0,1,\dots,\Big\lfloor\frac a2\Big\rfloor-1,\Big\lfloor\frac a2\Big\rfloor\Big\}:\sqrt{x(a-x)}\in\Bbb Z\}\Big|$$ and though likely non-...
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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-...
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Are the Gessel sequence integers composite for all $n\ge 3$?
The Gessel sequence is known for Ira Gessel's Lattice Path Conjecture of $2001$, which has been proved by Kauers, Koutschan and Zeilberger in $2009$ with the aid of a computer. Later, other proofs ...
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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 ...
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Does this sequence of ratios of digit sums have a limit?
I asked this question a few hours ago on MathStackExchange and there it received some attention but we still do not have a proof so I decided to ask it here also, in an unchanged form, and here it is:
...
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Identities for powers of functions based on generalization of Lagrange interpolation
Lagrange polynomial can be used to obtain an identity:
$$(k+t)^n = \sum_{i=0}^n (k+d_i)^n \prod_{\substack{j=0\\ j\not=i}}^n \frac{t-d_j}{d_i-d_j},$$
which holds for any integer $n>0$, any real ...
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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 \...
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Reference request for multiple free sequences
Erdos usually named a sequence of integers no one of which is divisible by any other as an $M$- sequence (M stands for "multiple-free") or primitive sequence.
For example it is easy to see that $\...
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Another integral that has a closed form involving finite series of $\zeta(2k+1)$'s. Could it be reflexive?
In the context of a series of questions here, here and here, about closed form expressions involving finite series of $\zeta(2k+1)$'s for certain integrals, I would like to raise another one:
$$f(n):=...
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Source coding lexicographic index of finite alphabet sequence with weight (partitions)
My goal is to determine the lexicographic index of an $M$-ary $n$-sequence $\mathbf{x}$ on the subset with an $M$-weight sum constraint:
$$S = \{ \mathbf{x} \in \{0, \ldots, M-1\}^n: \sum_{j=1}^n x_j =...
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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.
...
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In search of multiple expressions for a sequence
The sequence $a_n=\sum_{k=0}^n\binom{n}k^24^k$ is listed on OEIS along with a couple of combinatorial interpretations. What interested me at the moment is the plethora of binomial single-sums for the ...
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What are non-trivial facts about the sequence of averages of digits of an integer sequence?
Write $A_{10}(k)$ for the average of the base-10 digits of a positive integer $k$:
$A_{10}(k):=\tfrac{1}{L+1}(d_0+\dots+d_L)$, where $k=\sum_{i=0}^L d_i 10^i$ with $d_i\in\{0,\dots,9\}$
I wonder if ...
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Determining the asymptotic behavior of a sequence
I've encountered the following sequences
$$
a_k=2^{k+1}\sum_{j=0}^{k-1}a_{k-1-j}a_j,\;a_0=1
$$
$$
b_k=(k+1)\sum_{j=0}^{k-1}b_{k-1-j}b_j,\;b_0=1.
$$
I would like to have an estimate of the growth of ...
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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 ...
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Kindda-Perfect number: Is there a sequence of numbers which are equal to the sum of its proper divisors excluding itself as well as 1?
Perfect number is a positive integer that is equal to the sum of its proper divisors. The smallest perfect number is 6, which is the sum of 1, 2, and 3.
Is there a sequence of numbers which are equal ...
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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 ...
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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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Can the integers in an easily computable sequence free of prime numbers always be factored easily?
Call a sequence $(a_n)$ of positive integers easily computable
if there is a constant $C$ and an algorithm which computes $a_n$ from
$n$, $a_1, \dots, a_{n-1}$ and a finite number of integer ...
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Is every sequence that looks like an AP really an AP?
Caveat: I am not at all a number theorist, and I randomly came up with the following question while I was hiking. But I already asked two serious number theorists, and since they did not know the ...
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Counting Bipartitions
Numerical evidence suggests that $p_2(n) \geq n p(n)$ for large $n$.
Here $p(n)$ is the number of partitions of $n$, and $p_2(n)$ is the number of bipartitions of $n$, i.e., ordered pairs of ...
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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 $(1,2,3, \ldots, n)$...
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Does the divergent solution of this equation :$f'=e^{f^{-1}}$ of Gevrey type and could be Borel summation applied for it?
This question was asked here in MO by someone seeking for the solution of the functional -differential:$f'=e^{f^{-1}}$ not exactly an O.D.E, and again here seeking for the growth rate of it solution ...
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Terminology for "approximately convex" sequence of integers
I have a sequence of integers meeting the following inequality:
$u_n \leq \frac{u_{n-1}+u_{n+1}}{2} + \frac{1}{2}$. In other words, the sequence is "approximately convex", and the difference comes ...
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When do almost all these invariants of tensors vanish?
Let $A,B,C,D$ be $n$-dimensional vector spaces over a field $k$.
There is a natural homomorphism from the $mn^m$th tensor power $A^{\otimes (m n^m)} $ of $A$ to $k$ given by the determinant map $A^{\...
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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 ...
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What is the class of real sequences satisfying these conditions?
I'm interested in finding the class of the real sequences $u_{k}$, $k\in \mathbb{N^*}$ which satify the following conditions:
$\displaystyle \sum_{k=1}^{\infty}\frac{1}{u_{k}}$ diverges i.e $\...
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"flavored" equivalence classes of permutations
We say two permutations $\pi_1$ and $\pi_2$ in the symmetric group $\mathfrak{S}_n$ are $k$-equivalent, denoted
$\pi_1 \sim_k \pi_2$, if one can be
determined from the other after a finite number of ...
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are these polynomials or rationals functions?
Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by
$$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$
...
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Can you tie up these Laurent sequences?
Fix an integer $k\geq3$. Define the two families of sequences $\{x_n\}$ and $\{y_n\}$ according to the rules:
$$x_n=\frac{x_{n-1}^2+x_{n-2}^2+\cdots+x_{n-k+1}^2}{x_{n-k}} \qquad n\geq k$$
and
$$y_n=\...
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Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? [closed]
I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\...
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An interesting problem which I think only needs elementary number theory
A problem about elementary number theory
While writing my paper, I came across the following problem:
(all the discussion assume that $q$ is prime and $\alpha $ is a positive integer. ) We first ...
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Number Theory and p-Power-Partitioned Numbers
Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, we're going to define its digits-partition as the next set $D_{N} = \bigcup_{j=1}^{n}\bigcup_{k=1}^{p(a_{j})}\{(P_{k},j)\}$, where each pair $(...
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Coefficients for Powers of the Mittag-Leffler Function
Considering the one parameter Mittag-Leffler function,
$$E_{\alpha}(z)=\sum_{k=0}^\infty\frac{z^{k}}{\Gamma(\alpha k+1)}, \Re(\alpha)>0$$
Considering then the generating function for $E_\alpha(z^...
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GCD for two Cullen numbers
The $n$'th Cullen number is $C_n = n\cdot2^n+1$.
If $m$ and $n$ are natural numbers, what can one say about $\gcd(C_n,C_m)$, where $m$ and $n$ are different positive integers?
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Betweenness in permutations
Let us consider permutations $\pi$ on $\{1,\dots,n\}$ as sequences $\pi(1),\pi(2),\dots,\pi(n)$. For a permutation $\pi$ let $R(\pi)$ be the ternary relation with $(x,y,z)\in R(\pi)$ whenever element $...
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Number Theory and d-Self-Contained Numbers
Given any natural number $N = a_{n}a_{n-1}\ldots a_{1}$, let us associate to it the set $S_{N} = \bigcup_{j=1}^{n}\{(a_{j},j)\}$. We're going to define a d-self-contained number as any natural number ...
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Number of subsets that sum to $0$
Suppose you choose $n$ distinct random numbers from a contiguous subset of cardinality $f({\beta, n})$ with at least $f({\alpha_+, n})$ positive and at least $f({\alpha_-, n})$ negative values from a ...
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Repdigit numbers, which are sum of consecutive squares
Following up on this question,
https://math.stackexchange.com/questions/1788015/is-112122132142152162-1111-special/1788102?noredirect=1#comment3649733_1788102
is anything known about the sequence of ...
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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'...
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Arbitrarily many primes in a Fibonacci-type sequence
It is conjectured that the standard Fibonacci sequence contains infinitely many primes. While this is perhaps too difficult, I am wondering about the following simpler version:
Question. For any $K$, ...
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Integer Recursion Reference Request
I've run across the following recursion which at times seems very steady and predictable and at other times seems very chaotic.
Let $c_1, \dots c_k, b_0, m \in \mathbb{Z}$ with $b_0>m\ge 3$ and $...
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Why do the adjoint representations of three exceptional groups have the same first eight moments?
For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...
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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 ...
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