Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
23
votes
3
answers
2k
views
Zeroes of the random Fibonacci sequence
Let $X_n$ be the "random Fibonacci sequence," defined as follows:
$X_0 = 0, X_1 = 1$;
$X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips.
It is known ...
- 19.2k
69
votes
1
answer
4k
views
Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?
I've discovered what I believe is a quite remarkable sequence (A318970), defined by
$$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$
Here are the first four terms with their prime ...
- 34.5k
6
votes
3
answers
2k
views
What is the motivation and purpose of the Floretion group?
When searching through the Oeis, I came across something called a floretion. Based on the context, it seems to be some sort of algebraic structure. I googled it and found nothing that explained their ...
- 1,129
1
vote
1
answer
280
views
Sequences over finite fields
Let's we have finite field $F_q$ for some prime $q=2^M-1$.
I am looking for special sequence {$a_{i}$, $i \in {1,..,q-1}$},
($\{a_{1},...,a_{q-1}\}=F_q/\{0\}$) with the following properties:
$r_{1}=...
- 33
4
votes
1
answer
435
views
Quadratic progressions with very high prime density
In my previous MO question (see here), I solved the case for arithmetic progressions $f_k(x)=q_k x+1$. The solution is this:
The list of sequences $f_k(x)$, each one corresponding to a specific
$k$, ...
- 3,098
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
1
vote
1
answer
128
views
Bounds for the sequence $a(n,A)=n*a(\lfloor (1-A)n \rfloor,A)$
Related to this question and possibly the open problem
of the exponential time hypotheses.
Let $A$ be rational number, $0 < A < 1$.
For positive integer $n$, define the sequence
$a(1,A)=1$ and $(...
- 25.4k
49
votes
4
answers
4k
views
Strange (or stupid) arithmetic derivation
Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
- 3,004
14
votes
1
answer
835
views
Special configurations on a circle from a homological algebra problem
Here is the short version of the combinatorial problem:
Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...
- 26.5k
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
6
votes
0
answers
284
views
Is there a positive odd $n$ such that $\sigma(\sigma(n)) = \sigma(\sigma(n)-n)+\sigma(n)$?
Let $\sigma(n)$ denote the sum of the divisors of $n$. (https://oeis.org/A000203)
It is relatively easy to find numbers $n$ such that $f(g(n)) = g(f(n))$ where $f(n) = \sigma(n)$ and $g(n) = \sigma(n) ...
- 701
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,
$...
7
votes
0
answers
184
views
Upper bounds for a sequence of integers
Given $\alpha\geq0$ we consider the sequence
$$
C_k=k^\alpha\sum_{j=0}^{k-1}C_jC_{k-1-j}
$$
with $C_0=1$. I'm interested in upper bounds (in terms of $\alpha$) for such a sequence. I know that when $\...
- 843
-4
votes
1
answer
250
views
What are the patterns of the sequence of polynomials? [closed]
In my research, I obtained a sequence of polynomials (I am only able to compute the first 4 of them):
\begin{align}
& f(2) = 1+t, \\
& f(3) = 1+4t+3t^2, \\
& f(4) = 1+6t+12t^2+7t^3, \\
&...
- 6,211
3
votes
1
answer
138
views
Properties of a certain sequence
During research I came to the following sequence:
Let $\lambda>1$
and define $n_{k+1}=\text{IntergerPart}[\lambda\cdot n_k]$ where we assume that $n_0$ is sufficently large integer, so that the ...
- 589
2
votes
0
answers
176
views
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
6
votes
1
answer
240
views
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
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
2
votes
0
answers
327
views
Why can one compute the sum of divisors of $n$ without factoring $n$?
Question links to paper
which states:
$$
\sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1)
$$
where $\sigma(n)$ is the sum of divisors of $n$.
Another similar ...
- 25.4k
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
0
votes
0
answers
60
views
Reference request: Counting integer sequences in homogeneous linear recurrences
Are there references in the literature that deal with the probability of finding an integer sequence in a linear homogeneous recurrence with constant coefficients $ \in \mathbb{Z}$? (or provides a way ...
19
votes
2
answers
581
views
Sequences with 3 letters
For a positive integer $n$ I would like to construct long sequences consisting of 0, 1 and 2's such that for any two subsequences consisting of $n$ consecutive elements the number of 0's , 1's or 2'...
- 2,286
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 $$\frac{3a^...
19
votes
1
answer
1k
views
Is every sequence that looks like an AP really an AP?
Caveat: I am not at all a number theorist, and I randomly came up with the following question while I was hiking. But I already asked two serious number theorists, and since they did not know the ...
- 11.9k
2
votes
0
answers
137
views
Writing integers as sequences of products by 2 and integer divisions by 3
For any integer, we consider its decompositions into sequences of products by $2$ and integer division by $3$.
For instance:
$$
100 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \...
- 1,444
4
votes
0
answers
300
views
On $\sum_{k=1}^nk^3 = x^3 + y^3$ with $x,y \ge 1$
My question is related to https://oeis.org/A269839.
It is well-known that there are parametric families of solutions for cubes that are sums of consecutive cubes: https://arxiv.org/pdf/1603.08901.pdf. ...
- 701
7
votes
1
answer
386
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, ...
- 71
4
votes
1
answer
175
views
A binomial coefficient identity involving two parameters
In a recent calculation I obtain a result involving the following expression depending on two integers $n,m\geq 0$:
$$S(n,m):=\frac{(n+m+1)!}{n!m!}\sum_{l=0}^{n+m}\frac{1}{n+m-l+1}\sum_{\substack{j+k=...
- 1,942
7
votes
2
answers
428
views
Limit associated with complementary sequences
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_0b_{n-1}+a_1b_{n-2}$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
- 3,443
3
votes
0
answers
285
views
Catalan numbers, Pochhammer symbols, Stirling numbers of the second kind, and sums of aliquot parts
For integers $N\geq 1$ we define $$s(N)=\sigma(N)-N$$ the aliquot sum function, where $\sigma(N)=\sum_{1\leq d|N}d$ is the sum of divisors function.
Here $(x)_n$ is the Pochhammer symbol and ${a\...
16
votes
2
answers
1k
views
are these polynomials or rationals functions?
Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by
$$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$
...
- 43.2k
7
votes
0
answers
147
views
Factor-counting sequence
Define a non-negative integer sequence $\{\mathcal{F}_n\}$ as follows: start with 1 and, at each step, insert the number of entries already present in the sequence which are factors of the last one.
...
- 4,459
6
votes
1
answer
2k
views
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 ...
- 189
8
votes
0
answers
237
views
Sequences for which $\prod (1-z^n)^{a(n)}$ is a polynomial
This is mostly a reference request.
I'm working with complex coefficients, although all I have in mind have integer coefficients.
Let $a=(a(n))_{n\ge 1}$ be a sequence, say of integers (I have non-...
- 63.9k
0
votes
1
answer
379
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}} ...
- 3,443
0
votes
1
answer
144
views
Formally confirm a formula for a certain three-dimensional constrained integral over the unit cube
The result of the three-dimensional constrained integration (for the Hilbert-Schmidt two-qubit absolute separability probability) over the unit cube $[0,1]^3$
\begin{equation} \label{one}
\int_0^1 \...
- 723
18
votes
1
answer
607
views
Order of Conway's "look and say" recurrence
Let $L_n$ be the length of the $n$th term of Conway's "look and say"
sequence (https://oeis.org/A005341). The generating function $F(x)=
\sum_{n\geq 0}L_nx^n$ is a rational function, say $P(x)/Q(x)$ ...
- 50.8k
14
votes
5
answers
977
views
Is the sequence $a_n=c a_{n-1} - a_{n-2}$ always composite for $n > 5$?
Numerical evidence suggests the following.
For $c \in \mathbb{N}, c > 2$ define the sequence $a_n$ by
$a_0=0,a_1=1, \; a_n=c a_{n-1} - a_{n-2}$
For $ 5 < n < 500, \; 2 < c < 100$ there ...
- 25.4k
7
votes
0
answers
280
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]}$$ with $...
- 5,622
4
votes
1
answer
1k
views
The range of the Euler totient function and multiplication by 28
If $n$ is in the range of the Euler totient function, certain multiples of $n$ are likewise guaranteed to be totient values. The simplest nontrivial example of this is that, if $n$ is in the range of ...
- 3,655
1
vote
1
answer
176
views
The sequence $G(n,k)=G(n-2,k)+G(n,k-2)$
Background: The binomial coefficients $C(n,k)$ satisfy the recurrence
$C(n,k)=C(n-1,k)+C(n-1,k-1)$ and some terminating conditions, for
more information check here.
$C(n,k)$ doesn't appear to be ...
- 25.4k
0
votes
1
answer
296
views
Solutions of the equation $\psi(\sigma(n))=2n$, where $\sigma(n)$ is the sum of divisors function and $\psi(n)$ the Dedekind psi function
For integers $m\geq 1$ let $\sigma(m)$ the sum of divisors function $\sum_{1\leq d\mid m}d$ and let $\psi(m)$ the Dedekind psi function (as reference I add the Wikipedia Dedekind psi function), then ...
7
votes
0
answers
184
views
Some conjectural congruences involving Domb numbers
The Domb numbers are given by
$$D_n=\sum_{k=0}^n\binom{n}{k}^2\binom{2k}k\binom{2(n-k)}{n-k}\ \ \ (n=0,1,2,\ldots).$$
Such numbers have combinatorial interpretation, see, e.g., http://oeis.org/A002895....
- 15.6k
23
votes
5
answers
1k
views
Sequences with integral means
Let $S(n)$ be the sequence whose first element is $n$, and from then onward,
the next element is the smallest natural number ${\ge}1$ that ensures that the
mean of all the numbers in the sequence is ...
- 151k
4
votes
1
answer
245
views
Count weighted integer compositions
What is the asymptotic growth of the sequence
$$a_n:=\sum_{k\geq 0} 3^k c_{n,k},$$
as $n\rightarrow\infty$, where $c_{n,k}$ denotes the number of integer compositions of $n$ with exactly $k$ many 2s?
...
- 499
2
votes
2
answers
273
views
Alternating binomial-harmonic sum: evaluation request
Let $H_k=\sum_{j=1}^k\frac1j$ be the harmonic numbers.
QUESTION. Can you find an evaluation of the following sum?
$$\sum_{a=1}^b(-1)^a\binom{n}{b-a}\frac{H_{b-a}}a.$$
- 43.2k
15
votes
1
answer
475
views
Determinant of a matrix filled with elements of the Thue–Morse sequence
Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row with the first $n^2$ elements of the Thue–Morse sequence (with indexes from $0$ to $n^2-1$). Let $\mathcal D_n$ be ...
- 6,829
1
vote
0
answers
151
views
On smoothness and roughness of a number related to triangular numbers
Define $\triangle_n$ to be the $n$th triangular number.
Define $$M_n=(2\triangle_n-1)2\triangle_n(2\triangle_n+1)=2\triangle_n(4\triangle_n^2-1).$$
Define $(\ell,k)$-smough numbers to be numbers that ...
- 1,836
1
vote
2
answers
307
views
A question about integer triples
How can we generate all integer solutions of the equation
$$(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
139
views
(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