Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
2
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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
2
votes
2
answers
422
views
Why are attempts to define chaos with discrete states so scarce?
Interestingly, the theory of nested recurrence relations has been correlated with “discrete chaos” by Golomb (1991) and Tanny (1992).
And in literature, there are very few studies that have different ...
- 701
32
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0
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A question related to the Hofstadter–Conway \$10000 sequence
The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
- 701
1
vote
1
answer
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(Translation request) Hypotheses of the Blom-Fredberg bounds on denumerants?
I don't know Swedish and I'm not finding the article "G. Blom and C. E. Froberg, On money changing" translated into English... so I tried to read the original (Swedish) with the help of ...
- 13
6
votes
1
answer
240
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On the growth and bounds for a certain sequence of integers known as Bogotá numbers
A Bogotá number is a non-negative integer equal to some smaller number, or itself, times its digital product, i.e. the product of its digits. For example, 138 is a Bogotá number because 138 = 23 x (2 ...
- 3,717
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0
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My research paper involves computing additional terms of an existing OEIS sequence. Should I first amend the sequence or publish the results?
In the course of my research I computed terms of an existing OEIS sequence that are currently unknown. Having prepared my paper for publication, I am now faced with a (small) dilemma:
Do I first ...
- 1,962
7
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0
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280
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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 $...
- 5,622
4
votes
1
answer
175
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A binomial coefficient identity involving two parameters
In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$:
$$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=...
- 1,942
11
votes
3
answers
684
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Series and sequences in physical systems & closed form expressions
I gave a colloquium a while ago about physics inspiring recent developments in mathematics and as is almost borderline cliche in such talks, I mentioned the Fibonacci sequence with closed form ...
5
votes
3
answers
1k
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What is the asymptotic of the irregular blue curve? Is it $(8x)^{1/2}$ or is it something else?
From Terry Tao's post here there is the statement:
"Conversely, if one can somehow establish a bound of the form
$$\displaystyle \sum_{n \leq x} \Lambda(n) = x + O( x^{1/2+\epsilon} ) \tag{1}$$
...
- 1,183
3
votes
2
answers
285
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 ...
- 5,590
0
votes
0
answers
54
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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 ...
- 11
4
votes
0
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97
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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 ...
- 25.4k
3
votes
1
answer
173
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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 ...
- 128k
1
vote
0
answers
73
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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
115
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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.
$$
This polynomial is related to a family of two-...
- 29
4
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0
answers
178
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Primitive roots modulo primes related to Fibonacci numbers or Lucas numbers
The Fibonacci numbers $F_0,F_1,F_2,\ldots$ and the Lucas numbers $L_0,L_1,L_2,\ldots$ are given by
$$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{...
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0
votes
1
answer
61
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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
$$0\...
- 1,836
15
votes
1
answer
1k
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Arithmetic progressions in stopping time of Collatz sequences
Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.
We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
- 259
1
vote
2
answers
307
views
A question about integer triples
How can we generate all integer solutions of the equation
$$(qr+rp+pq)(x^2+y^2+z^2) = (p^2+q^2+r^2)(yz+zx+xy),$$
given that $p,q,r$ are integers?
Clearly if any one of $(x,y,z), (x,z,y), (y,z,x), (...
- 3,443
1
vote
1
answer
334
views
Are there infinitely many primes of the form $\frac{3a^2-a}{2}+b^4$?
I was inspired from a theorem due to Iwaniec and Friedlander, see [1], to ask the following conjecuture involving integers.
Conjecture. There are infinitely many prime numbers of the form $$\frac{3a^...
0
votes
1
answer
296
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
247
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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 ...
- 3,717
1
vote
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
5
votes
2
answers
1k
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,
$...
19
votes
2
answers
2k
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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 ...
- 193
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
0
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1
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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
285
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Catalan numbers, Pochhammer symbols, Stirling numbers of the second kind, and sums of aliquot parts
For integers $N\geq 1$ we define $$s(N)=\sigma(N)-N$$ the aliquot sum function, where $\sigma(N)=\sum_{1\leq d|N}d$ is the sum of divisors function.
Here $(x)_n$ is the Pochhammer symbol and ${a\...
3
votes
0
answers
127
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 ...
- 18.7k
0
votes
1
answer
95
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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 ...
- 9
2
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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. ...
4
votes
2
answers
593
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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 ...
- 87
0
votes
1
answer
149
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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
393
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)\...
- 153
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
135
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, ...
- 3,153
0
votes
0
answers
86
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Polynomials of integer coefficients that evaluated at Mersenne or Fermat numbers produce 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 to get prime constellations or statements related to primality tests for these ...
8
votes
0
answers
145
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
177
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:
$$\begin{array}{ll} \text{minimize} &...
- 1,183
4
votes
1
answer
245
views
Count weighted integer compositions
What is the asymptotic growth of the sequence
$$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$
as $n\rightarrow\infty$, where $c_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s?
...
- 499
1
vote
0
answers
94
views
Family of polytopes whose measure respects multiplication?
Is there a family $\mathcal{P}$ of integral polytopes and a polytope product $\star$ such that for every $n\in\mathbb N_{>1}$ $\exists p\in\mathcal{P}:vol(p)=n$ and
$\forall q\in\mathcal{P}\...
- 13.9k
0
votes
0
answers
248
views
Divisibility Properties of Pisano Periods
Let $(F_n)$ the Fibonacci sequence and $\pi(m)$ the Pisano period of $m$ (i.e., the smallest period of $F_n \pmod{m}$). There are many proved results about $\pi(m)$. For example, it is known that $\pi(...
- 21
0
votes
0
answers
72
views
Rewriting a set of integers to get rid of repetition but keeping subset sum ordering
Say, I have a set of 6 +ve integers sorted in ascending order:
$A = \{2,4,4,4,5,7\}$
Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them:
$\therefore B= ...
- 21
0
votes
1
answer
62
views
Mapping naturals to pairs of naturals and viceversa [closed]
I can't find much on the internet about this, but apparently vectors of naturals are called hyperscalars. It's not hard to bijectively map naturals to 2D hyperscalars and with that to prove that any-...
- 13
2
votes
2
answers
273
views
Alternating binomial-harmonic sum: evaluation request
Let $H_k=\sum_{j=1}^k\frac1j$ be the harmonic numbers.
QUESTION. Can you find an evaluation of the following sum?
$$\sum_{a=1}^b(-1)^a\binom{n}{b-a}\frac{H_{b-a}}a.$$
- 43.2k
9
votes
0
answers
304
views
Symmetric function transition matrix and a non-conjecture by Clifford and Stanley
Consider the transition matrix $R = \left(R_{\lambda,\mu}\right)$, defined by
$$
p_\lambda = \sum_{\mu} R_{\lambda\mu}m_\mu ,
$$
between the power-sum and the monomial basis of the ring of symmetric ...
- 15.8k
12
votes
1
answer
634
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 ...
- 6,829
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
10
votes
1
answer
589
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 ...
- 151k