Questions tagged [integer-sequences]

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

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5
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0answers
122 views

A recursion which defines polynomials with integer coefficients?

Let $[n]=1+q+\dots+q^{n-1}$ and $u(n)=\prod_{j=1}^n \gcd([j],[n])$. Define $$r(n)=\sum_{d|n,d>1}{(-1)^d \frac{u(n)}{du(\frac{n}{d})^d}r\Big(\frac{n}{d}\Big)^d}+\frac{(1-q)^{n-1}u(n)}{n[n]}$$ with $...
10
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0answers
226 views

Let $a(n) = a(\pi(n)) + a(n-\pi(n))$ with $a(1) = a(2) = 1$. What is $\lim_{n\to \infty} \frac{a(n)}{n}$?

My question is related to https://oeis.org/A316434. Let $$a(n) = a(\pi(n)) + a(n-\pi(n))$$ with $a(1) = a(2) = 1$, where $\pi(n)$ is the prime-counting function. Does the following limit exist? $$\...
4
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1answer
112 views

A binomial coefficient identity involving two parameters

In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$: $$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=...
11
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3answers
606 views

Series and sequences in physical systems & closed form expressions

I gave a colloquium a while ago about physics inspiring recent developments in mathematics and as is almost borderline cliche in such talks, I mentioned the Fibonacci sequence with closed form ...
7
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2answers
557 views

What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?

From Terry Tao's post here there is the statement: "Conversely, if one can somehow establish a bound of the form $$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$ ...
2
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1answer
107 views

Distinct distances between adjacent equal elements

Let's call a sequence $a_1, \ldots, a_n$ suitable if for any positive integer $d$ there is at most one index $i$ such that $a_i = a_{i + d}$ and all elements $a_{i + 1}, \ldots, a_{i + d - 1}$ are not ...
0
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0answers
43 views

A distribution of maximum of sums if add to the minimal

Consider a vector of $n$ integer variables with initial values of 0. Each step we take random $w_i\thicksim NB(q, l)$ (independent randon values with the same negative binomial distribution) and add ...
4
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0answers
89 views

When is $\lfloor C^n \rfloor \mod b$ efficiently computable?

For real irrational $C > 1 $ and natural $n,b$, define $a(C,n,b)=\lfloor C^n \rfloor \mod b$ Q1 For which $C,b$ is $a(C,n,b)$ computable in time polynomial in $\log{n}$? Searching in OEIS ...
3
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1answer
95 views

How many flips of a fair coin are needed to get at least one run of at least $k$ consecutive heads with probability $\ge1/2$?

The following question was asked today: How many flips $n$ of a fair coin are needed to get at least one run of at least $k$ consecutive heads with probability $P_{k,n}\ge1/2$? The question was ...
0
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0answers
37 views

Sum of a two dimensional arithmetico-geometric suite

I am trying to compute the first column $P_{k,1}^n$ of the power of matrix $P^n$ where $P$ is a lower bidiagonal matrix with terms : $$P_{i,j} = \left\{\begin{array}{cc} i\alpha & \text{if }i=j,\\ ...
1
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0answers
41 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 ...
0
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0answers
100 views

Roots of a family of 4-parameter polynomials

Let $k, \ell, p$ and $q$ be positive integers, with $q>p>1$ and $\gcd(p,q)=1$. Let $f(x)$ the polynomial given by $$ f(x)=x^q-kx^{q-p}-\ell. $$ This polynomial is related to a family of two-...
4
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86 views

Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers

The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by $$F_0=0,\ F_1=1,\ \text{and}\ F_{n+1}=F_n+F_{n-1}\ (n=1,2,3,\ldots)$$ and $$L_0=2,\ L_1=1,\ \text{...
0
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1answer
58 views

Ordered $m$-tuples with fixed number of changes

Given $1\leq k\leq m$, $2\leq d\leq c i\ln i$ and $2\leq i\leq c'\ln(mi\ln i)$ at some $c,c'>0$ how many sequences (lower and upper bounds) are of form $$z_1,\dots,z_m$$ on the condition that $$0\...
14
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1answer
513 views

Arithmetic progressions in stopping time of Collatz sequences

Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern. We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
1
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2answers
283 views

A question about integer triples

How can we generate all integer solutions of the equation $$(qr+rp+pq)(x^2+y^2+z^2) = (p^2+q^2+r^2)(yz+zx+xy),$$ given that $p,q,r$ are integers? Clearly if any one of $(x,y,z), (x,z,y), (y,z,x), (...
1
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1answer
203 views

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^...
1
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1answer
149 views

Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function

For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...
3
votes
1answer
168 views

A special type of Langford pairing

A Langford pairing is a permutation of the sequence of 2n numbers 1, 1, 2, 2, ..., n, n in which the two 1s are one unit apart, the two 2s are two units apart, and more generally the two copies of ...
1
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1answer
162 views

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 ...
5
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2answers
860 views

Proof that $3^ns + \sum_{k=0}^{n-1} 3^{n-k-1}2^{a_k}=2^m.$

How would I go about proving the following: For any odd positive integer $s$, there exists a sequence of nonnegative integers $( a_0, a_1, \cdots, a_{n-1})$ and a nonnegative integer $m$ such that, $...
18
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2answers
2k views

A finite alternating sum

We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is: $$ S_n = \sum_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j $$ We have observed numerically that ...
1
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1answer
87 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 $\mathbb{N}$ ...
1
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1answer
102 views

A problem inspired in the definition of tau numbers and a divisibility relationship related to powers of two

It is (I assume that this easy fact is well-known) obvious that an integer $n>1$ is a power of two $n=2^{\alpha}$, where $\alpha\geq 1$ is integer, if an only if $n$ satisfies the divisibility ...
3
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0answers
125 views

Is there a name for this operation on integer functions?

Suppose $f$ and $g$ are functions from $\mathbb N^+$ to itself. I want to consider the function $f^g$, where $f^g(n) = f \circ \dots \circ f(n)$, where composition is done $g(n)$-many times. Note ...
0
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1answer
84 views

The growth of a sequence related to Liouville numbers [closed]

I am doing a work on Liouville numbers. The Liouville constant $\ell=\sum_{k\geq 0}10^{-k!}$ has its approximation by rational numbers related to the fact that for $v_n=n!$, then $v_{n+1}/v_n$ tends ...
1
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0answers
60 views

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. ...
3
votes
1answer
146 views

Squares in Lucas sequences

Good night, everyone! According to a celebrated result by J. H. Cohn, the only perfect squares in the Fibonacci sequence are $F_{0}=0$, $F_{1}=F_{2}=1$, and $F_{12}=144$. It is also known that the ...
0
votes
1answer
86 views

Asymptotic of $\sum_{k=1}^n \operatorname{rad}(k!)$ and similar deductions

We denote for integers $m>1$ the product of the distinct prime numbers dividing $m$ as $$\operatorname{rad}(m)=\prod_{\substack{p\mid m\\p\text{ prime}}}p,$$ with the definition $\operatorname{rad}(...
5
votes
2answers
360 views

What is this sequence counting?

While solving (a system of) a system of linear equations level-by-level recursively, I am finding some redundant equations for level $n\geq5$. The reason why the redundancies arise is because $P(n)\...
1
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0answers
19 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
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0answers
122 views

Permutation of a sequence, such that $y_i+y_{i+1}$ are all distinct

The sequence $x_1, x_2, ..., x_n$ of positive integers contains at least $\frac {2n}{3}+1$ distinct numbers and each of them appears at most three times. How to prove that there is a permutation $y_1, ...
0
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0answers
48 views

Attempt to set a conjecture concerning polynomials of integer coefficients, Mersenne or Fermat numbers, and square-free integers

Mersenne numbers $M_n=2^n-1$ and Fermat numbers $F_n=2^{2^n}+1$ draw the attention of professional mathematicians and amateurs to get prime constellations* from the definition of these sequences or ...
7
votes
0answers
119 views

Minimum length of sequence such that every integer from 1 to n can be achieved as the sum of some contiguous subsequence

This question literally came to me in a fever dream last night, and it's frustrating me to no end. I'll try to explain it as best I can, but there may not be a satisfying answer; the best outcome ...
1
vote
0answers
103 views

Prove that these linear programming problems are bounded by $O(k^{1/2})$ [closed]

The expanded partial sums of the Möbius inverse of the Harmonic numbers have two out of three properties in common with this set of linear programming problems: $$\begin{array}{ll} \text{minimize} &...
4
votes
1answer
175 views

Count weighted integer compositions

What is the asymptotic growth of the sequence $$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$ as $n\rightarrow\infty$, where $c_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s? ...
1
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0answers
90 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}\...
0
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0answers
108 views

Divisibility Properties of Pisano Periods

Let $(F_n)$ the Fibonacci sequence and $\pi(m)$ the Pisano period of $m$ (i.e., the smallest period of $F_n \pmod{m}$). There are many proved results about $\pi(m)$. For example, it is known that $\pi(...
0
votes
0answers
46 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= ...
0
votes
1answer
51 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-...
2
votes
2answers
233 views

Alternating binomial-harmonic sum: evaluation request

Let $H_k=\sum_{j=1}^k\frac1j$ be the harmonic numbers. QUESTION. Can you find an evaluation of the following sum? $$\sum_{a=1}^b(-1)^a\binom{n}{b-a}\frac{H_{b-a}}a.$$
9
votes
0answers
250 views

Symmetric function transition matrix and a non-conjecture by Clifford and Stanley

Consider the transition matrix $R = \left(R_{\lambda,\mu}\right)$, defined by $$ p_\lambda = \sum_{\mu} R_{\lambda\mu}m_\mu , $$ between the power-sum and the monomial basis of the ring of symmetric ...
11
votes
1answer
390 views

Integrals of power towers

Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
4
votes
0answers
151 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 ...
8
votes
1answer
482 views

XOR-free sets: Maximum density?

It is known that sum-free subsets of $\mathbb{N}$ can have natural density at most $\frac{1}{2}$. This density is achieved by the odd numbers: the sum of two odd numbers is even. I ask now a similar ...
25
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0answers
429 views

A sequence potentially consisting of only integers

I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences. Consider the sequence defined by $$b_n = \frac{(...
13
votes
2answers
2k views

Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$

Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers $$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$ As $T_x=\binom{x+1}2$, Gauss' triangular number ...
6
votes
1answer
314 views

Closed form expression for a recursion relation with binomial coefficients

I am interested in the following sequence: $$ T_n = \sum\limits^{n-1}_{k=0} \begin{pmatrix} n \\ k \end{pmatrix} T_{k}, \ \ \ \ T_0 = C \in \mathbb{N} $$ I would like to express it as a function of n, ...
2
votes
1answer
213 views

Numbers that are the sum of 2 distinct nonzero squares in exactly 1 way [closed]

I need to emulate this sequence for a program: http://oeis.org/A025302 Stuff that I've taken into account: After finding the prime divisors of a number. I take any divisor as p and apply the ...
0
votes
1answer
354 views

A possible surprise involving Euler's constant $e$ [closed]

Let \begin{align*} c_n &= n!\left(e-\sum_{k=0}^n \frac{1}{k!}\right) \\ \\ u_n &= \bigg\lfloor{\frac{1}{c_n} \bigg\rfloor} \\ \\ v_n &= \bigg\lfloor{\frac{1}{1/c_n-\lfloor{u_n} \rfloor}} ...