# Questions tagged [integer-sequences]

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

194
questions

**5**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**3**answers

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**

votes

**2**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**0**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**2**answers

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**

vote

**1**answer

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**

vote

**1**answer

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

**1**answer

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**

vote

**1**answer

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**

votes

**2**answers

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**

votes

**2**answers

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**

vote

**1**answer

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**

vote

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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**

vote

**0**answers

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**

votes

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**0**answers

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

**1**answer

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**

vote

**0**answers

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**

votes

**0**answers

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

**0**answers

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

**1**answer

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

**2**answers

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

**0**answers

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

**1**answer

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

**0**answers

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

**1**answer

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**

votes

**0**answers

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

**2**answers

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

**1**answer

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

**1**answer

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

**1**answer

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}} ...