Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
2
votes
1
answer
196
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Guess (or upper bound) the general formula for a double sequence
Let $t,s \geq 0$ be integers. We have the following recursive formula:
$$f(t+1,s) = f(t,s) + f(t,s-1) + \sum_{0\leq a,b,c \leq h(t):\\a+b+c = s-1}f(t,a)f(t,b)f(t,c),$$ where
$$h(t) = \frac{1}{2}3^t -\...
- 515
41
votes
1
answer
1k
views
Mod sequences that seem to become constant; and the number 316
Define a "mod sequence" of nonnegative integers
based on one start parameter $s$, its first term,
as follows.
$A(s)=(a_1,a_2,\ldots,a_n,\ldots)$
with $a_1 = s$
and
$$ a_n = \left(\sum_{k=1}^{n-1} a_k \...
- 151k
1
vote
0
answers
73
views
On a type of equations that involve certain multiplicative functions and polynomials, in relation to their number of solutions
Past weekend I was interested in the sequence A058891 from the On-Line Encyclopedia of Integer Sequences, from this, inspired by the equation due to Benoit Cloitre (2002) that shows the comments, I ...
14
votes
1
answer
4k
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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
0
answers
161
views
Consecutive integers each of which has a large prime factor
There are many results about consecutive integers all having small prime factors. But what about consecutive integers each of which has a large prime factor?
More precisely, let $P(n)$ be the ...
- 341
-4
votes
1
answer
178
views
Covering system of congruences with specific properties?
A family of residue classes $a_i (\bmod n_i)$ with $2\leq n_1\leq\cdots\leq n_r$, ($r\geq2$) is called a covering system of congruences if every integer belongs to at least one of the residue classes, ...
- 841
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
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
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
0
answers
94
views
Family of polytopes whose measure respects multiplication?
Is there a family $\mathcal{P}$ of integral polytopes and a polytope product $\star$ such that for every $n\in\mathbb N_{>1}$ $\exists p\in\mathcal{P}:vol(p)=n$ and
$\forall q\in\mathcal{P}\...
- 13.9k
3
votes
2
answers
203
views
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 ...
- 843
4
votes
0
answers
156
views
Inequalities about tripling and doubling sumsets
Let $A$ be a set of vectors in $\mathbb Z^d$ who $\mathbb R$-span is the whole $\mathbb R^d$. Let $s_i(A)$ denote the size of $A+A+\dots A$ ($i$ times). I am interested in the following:
Question 1:...
- 30.6k
4
votes
0
answers
206
views
Generating a Penrose tessellation around a given tile
Given a starting Penrose tile, I need to build a "spiraling" tessellation around it.
The following picture illustrates the request:
In this example, the starting tile is a "thin rhombus" (the pink ...
- 145
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
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
5
votes
0
answers
317
views
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}...
- 12.3k
1
vote
0
answers
223
views
Does each prime $p>3$ have a quadratic nonresidue which is a Mersenne number?
Recall that the Mersenne numbers are those integers $M_p=2^p-1$ with $p$ prime.
QUESTION: Is it true that for each prime $p>3$ there is a Mersenne number which is a quadratic nonresidue modulo $p$?...
- 15.6k
3
votes
1
answer
330
views
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 ...
- 133
8
votes
4
answers
1k
views
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
3
votes
1
answer
344
views
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:
$$f(n):=...
- 4,169
0
votes
1
answer
62
views
Mapping naturals to pairs of naturals and viceversa [closed]
I can't find much on the internet about this, but apparently vectors of naturals are called hyperscalars. It's not hard to bijectively map naturals to 2D hyperscalars and with that to prove that any-...
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-5
votes
1
answer
454
views
Is there a fixed integer $n$ for which the difference :$\pi^n-\ e ^n$ is integer number? [closed]
I'm interested knowing more about nature of $\pi$ and $\ e$ since they are independent algebraically.
In this question I'm interested to know if there exist a integer $n$ for which the difference $\...
0
votes
0
answers
72
views
Rewriting a set of integers to get rid of repetition but keeping subset sum ordering
Say, I have a set of 6 +ve integers sorted in ascending order:
$A = \{2,4,4,4,5,7\}$
Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them:
$\therefore B= ...
- 21
3
votes
1
answer
447
views
An interesting problem which I think only needs elementary number theory
A problem about elementary number theory
While writing my paper, I came across the following problem:
(all the discussion assume that $q$ is prime and $\alpha $ is a positive integer. ) We first ...
- 363
4
votes
0
answers
302
views
Identities for powers of functions based on generalization of Lagrange interpolation
Lagrange polynomial can be used to obtain an identity:
$$(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},$$
which holds for any integer $n>0$, any real ...
- 34.5k
0
votes
0
answers
88
views
Infinite difference length of integer subsets
Let $A$ be a set of integers. In our recent researches, we've faced to the following property and definition:
We say $A$ has infinite difference length, if
(a) For every integer $n$ there exist a ...
- 995
-1
votes
1
answer
72
views
Create approximations of finite integer sequence
Given a function of real numbers f(x), I can create approximations to arbitrary precision using Taylor polynomials.
Is there something equivalent in the discrete case when I have a sequence of ...
- 103
0
votes
1
answer
133
views
Elementary description to count of perfect squares - I
Is there an elementary description of $$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|$$ and though likely non-...
- 13.9k
2
votes
0
answers
120
views
Sieving the values of an arithmetic sequence which is infinitely many times $1$
I have a sequence of positive integers $a_n$ which assumes infinitely many times the value $1$. I want to estimate the cardinality of the following set:
$$\#\{n\leq x : a_n>1 \text{ and } (a_n, \...
4
votes
0
answers
105
views
Closed form for integer series from enumerative geometry problem?
Is there a closed form for the following integer sequence:
$$
1,6,145,8806,830622,100317140,14342519633,2325250316950,...
$$
This is the degree of the $2n$-th power of the Schubert class $\sigma_{2,...
- 17.4k
2
votes
1
answer
301
views
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 ...
user76479
9
votes
0
answers
398
views
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^{\...
- 149k
2
votes
3
answers
284
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 ...
- 747
0
votes
1
answer
310
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 on ...
- 151
4
votes
1
answer
253
views
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 $...
- 205
0
votes
1
answer
104
views
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+...
- 13.9k
2
votes
1
answer
238
views
"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 ...
- 43.2k
2
votes
1
answer
143
views
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 ...
- 21
14
votes
1
answer
427
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 ...
- 37.7k
6
votes
1
answer
450
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, ...
- 151k
0
votes
0
answers
540
views
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 ...
user89024
1
vote
0
answers
28
views
Asymptotic size for the number of terms not exceeding $n$ in the class $r$ for a classification of the type Erdös-Selfridge for square-free integers
It is possible to define a classification similar than the Erdös-Selfridge classification of primes for different sequences. Please ee [1], section A18 and the references cited in this book. Because ...
3
votes
0
answers
223
views
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 ...
user99666
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
2
votes
1
answer
380
views
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 ...
- 277
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
5
votes
0
answers
176
views
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 ...
- 19.6k
2
votes
1
answer
214
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 b_{k+1}=2b_k^...
- 621
2
votes
0
answers
154
views
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
votes
1
answer
154
views
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?
- 21