Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
6
votes
2
answers
741
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Shifting an irrational binary sequence
Let $\newcommand{\tn}{\{0,1\}^\mathbb{N}}\tn$ be the collection of all infinite binary sequences. For $s\in\tn$ and $k\in\mathbb{N}$ let the left-shift of $s$ by $k$ positions, $\ell_k(s)\in \tn$, be ...
- 48.9k
1
vote
0
answers
104
views
Curious congruences modulo $4$ involving primes
We define
$$S(n)=\sum_{a=2+(n\pmod 2)}^{n-2}
\sharp(\{j,1\leq j<n \pmod{a},(a,j)=1\})\ .$$
(Searching the OEIS yielded no results.)
For $n>2$ we have the following experimental observations (...
- 17.9k
2
votes
1
answer
277
views
Does the Apéry-like sequence $A_n=(n!)^2\sum_{k=0}^n { \rho \choose k}^2 { \rho+n-k \choose n-k}, \rho=e^{2 \pi i/3}$ change signs infinitely often?
This is an integer sequence OEIS sequence A217703.
It satisfies an order 3 recurrence which is the constant term $A_n=u_n(0)$ of a three term recurrent sequence of polynomial defined by
$$u_0(x)=1,u_1(...
- 1,362
6
votes
1
answer
315
views
Integral points of polynomials - a Furstenburg-type "topology" on $\mathbb{Z}$
Given $S \subseteq \mathbb{C}$, define $\displaystyle \mathfrak{c}(S) = \bigcap_{p(x) \in \mathbb{C}[x] \wedge p(S) \subseteq \mathbb{Z}}p^{-1}(\mathbb{Z}) \supseteq S$ ("the integral points ...
- 1,543
6
votes
1
answer
173
views
$\omega$-de-Bruijn sequences
Let $\omega$ denote the set of non-negative integers. For which integers $n>1$ is there a sequence $b_n: \omega\to\omega$ with the following property?
Whenever $v\in\omega^n$ there is a unique $...
- 48.9k
10
votes
0
answers
252
views
Permutation of positive integers
Let $a_n$ be a sequence such that $a_1=1$ and for each $n \geq 1$ $a_{n+1}$ is the smallest positive integer distinct from $a_1,a_2,...,a_n$ such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$....
- 3,153
3
votes
1
answer
435
views
What is the connection between these three methods of generating this sequence?
I was recently looking at this problem: “There are a number of balls in a jar, some of them red, some of them white. The odds of picking two at random and both balls being red is 1/2. How many of the ...
- 39
4
votes
1
answer
241
views
Do there exist prime numbers of the form $n \cdot 2^n + 1$, when $n \in \mathbb{N}$ and $n > 1$?
Recently, I was studying prime sequences of the form $k \cdot 2^n + 1$, and I noticed that primes of the form $n \cdot 2^n + 1$ almost do not exist, except for the $n = 1$ case.
Are there other prime ...
- 125
1
vote
0
answers
153
views
A new Conjecture at OEIS sequence A376842
Here is a new conjecture of mine from the appendix of an unpublished manuscript currently under review.
Let $b \in \mathbb{Z}^+$ and assume that $n$ is an integer greater than $1$ and not a multiple ...
- 1,451
1
vote
0
answers
82
views
Generating functions related to generating function of Catalan numbers
Let $C_n$ be A000108 (i.e., Catalan numbers). Here generating function is $C(x)$ such that
$$
C(x) = \frac{1-\sqrt{1-4x}}{2x}.
$$
Let $a(n)$ be an integer sequence with generating function $A(x)$ such ...
- 4,948
6
votes
0
answers
171
views
An inequality involving integer partitions
For integers $n\ge k\ge0$, let $p(n,k)$ denote the number of ways to write $n$ as a sum of $k$ positive integers (repetition allowed). For example, $p(6,3)=3$ since
$$6=1+1+4=1+2+3=2+2+2.$$
QUESTION. ...
- 15.6k
5
votes
1
answer
172
views
On vanishing of $p$-adic logarithms
Might be related to Wieferich primes.
Let $p$ be odd prime and define the Fermat quotient
$$F(n)=\frac{(2^{n-1} -1)}{n} \mod n=\frac{(2^{n-1} \bmod n^2 )-1}{n}$$
For integer $b$ let $L_p(b)$ be the $p$...
- 25.4k
6
votes
1
answer
284
views
Integer sequences with a periodic pattern
Let $A$ and $B$ be two different integers. Let $S$ be a finite integer sequence with exactly $n_A$ $A$s and $n_B$ $B$s. By repeating $S$ infinitely many times we obtain an infinite integer sequence $P$...
- 93
1
vote
0
answers
84
views
Coarse well-distributedness/equidistribution of Pell sequence prefixes
I am interested in the distributedness or "mixing" behavior of certain
linear recurrences modulo powers of $2$.
In particular, consider the Pell sequence (https://oeis.org/A000129),
modulo $...
- 11
5
votes
0
answers
183
views
On the polynomials $\sum_{k=0}^n\binom{n+k}k^m q^k$
A sequence of polynomials
$$P_0(q),\ P_1(q),\ P_2(q),\ \ldots$$
with real coefficients is called $q$-log-convex if for each $n=1,2,3,\ldots$ every coefficient of the polynomial $P_{n+1}(q)P_{n-1}(q)-...
- 15.6k
16
votes
2
answers
2k
views
Does this sequence ever end?
This may help: A080670 A195265
Define $f(n)$ as this:
Take a number $n$, and split it into its prime composition using $^$ and $×$. Now remove all $^$ and $×$, you get a new number, this is $f(n)$ (...
- 467
2
votes
0
answers
76
views
upper and lower bounds on rowlands sequence
rowlands sequence is defined as follows
\begin{equation}
a_{n}=a_{n-1} + b_{n}
\end{equation}
where $b_{n} = gcd(a_{n-1}, n)$ for $n>h$
it originates from E. Rowlands 2008 paper "A Natural ...
6
votes
2
answers
389
views
Conjectured Somos-like closed form of recurrences with polynomial coefficients
From Our short paper
For polynomial $F$ with integer coefficients, define the recurrence
$f(n)=F(n,f(n-1),f(n-2),...,f(n-d))$. We conjecture that
$f(n)$ satisfy Somos like sequence
$f(n)=\frac{G(f(n-1)...
- 25.4k
2
votes
1
answer
217
views
Number of distinct higher dimensional integer partitions
By a distinct partition, I mean a partition into distinct parts, i.e., $10 = 5+4+1$ is one, but $10=6+2+2$ is not. The number of distinct partitions of $k$ all whose parts are at most $n$ is given by ...
- 281
1
vote
0
answers
60
views
On the parity of $(2^{\varphi(n)}-1) \bmod{n^2}$
For odd integer $n$ define the function
$$ J(n)=(2^{\varphi(n)}-1) \bmod{n^2}$$
$J(n)$ is integer in $[0,n^2-1]$ and it is divisible by $n$.
Integer $n$ is Wieferich number
iff $J(n)=0$ and if $n$ is ...
- 25.4k
3
votes
1
answer
329
views
Nonexistence of short integer program sequence which generates squares
Is there a way to show within an integer program with constant number of variables and constraints of length $poly(\log B)$ (say length $\leq10^{1000000}\log B$), it is not possible for a variable to ...
- 13.9k
2
votes
0
answers
62
views
Algorithm for main diagonal of integer coefficients associated with Schroeder numbers
Let $T_q(n, k)$ be an integer table such that
$$T_q(n, k) = \begin{cases}
1 & \textrm{if } n = 0 \vee k = 0 \\
qT_q(n-1, n-1) + T_q(n, n-1) & \textrm{if } n = k > 0 \\
T_q(n, k-1) + T_q(n-1,...
- 4,948
12
votes
4
answers
1k
views
Six consecutive positive integers with certain shape
Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ?
If they exist, one of those six integers A will be the product of 2 and a square of ...
- 321
2
votes
1
answer
216
views
Simplification of the closed form for the A329369
Let $s(n,k)$ be a (signed) Stirling number of the first kind.
Let ${n \brace k}$ be a Stirling number of the second kind.
Let
$$
f(n,m,i) = (-1)^{m-i+1}\sum\limits_{j=i}^{m+1}j^n s(j,i) {m+1 \brace ...
- 4,948
1
vote
0
answers
168
views
Integer coefficients and integrals
Let $a(n,p,q)$ be the family of integer sequences such that exponential generating functions for it satisfy
$$
A_1(x)=\exp\left(x + p\int\int (A_1(x))^q \, dx \, dx\right).
$$
Let $b(n,p,q)$ be the ...
- 4,948
6
votes
1
answer
367
views
On A057985 and A287066
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0 \to 01$, $1 \to 12$, $2 \to 0$).
Let $b(n)$ be A287066 (i.e., start with $1$ and repeatedly substitute: $0 \to 01$, $1 \to 12$...
- 4,948
1
vote
0
answers
106
views
Simpler recursion for the A358612
Let $T(n,k)$ be an integer coefficients (A358612) such that
$$
T(2n+1, k) = kT(n, k) + T(n, k-1), \\
T(2n, k) = kT(n, k) + T(n, k-1) - \frac{T(2n, k-1) + T(n, k-1)}{k-1}, \\
T(n, 1) = T(0, 2) = 1
$$
...
- 4,948
1
vote
1
answer
221
views
Correctness of the algorithm for the A329369, A347205 and related sequences
Let $a(n)$ be A347205. It is enough for us to know that
$$
a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^jk), \\
a(0) = 1
$$
Let $b(n)$ be A329369. It is enough for us to know that
$$
b(2^m(2k+1)) = \sum\...
- 4,948
7
votes
1
answer
527
views
Suitable closed form for the A079501
Let $a(n)$ be A079501 (i.e., number of compositions of the integer $n$ with strictly smallest part in the first position).
The sequence begins with
$$
1, 1, 2, 2, 4, 5, 8, 12, 19, 28, 45, 70, 110, ...
- 4,948
6
votes
1
answer
393
views
Test for pair of odd primes $(p, 2p^2-1)$
Let $a(n)$ be A106483 (i.e., primes $p$ such that $2p^2-1$ is also prime).
Let $b(n)$ be an integer sequence such that $b(n) = B$ after the whole transformation where we start with $A = n$, $B = 1$, $...
- 4,948
4
votes
1
answer
148
views
Closed form for the A110501 (unsigned Genocchi numbers (of first kind) of even index)
Let $a(n)$ be A110501 (i.e., unsigned Genocchi numbers (of first kind) of even index). Here
$$
a(n) = \sum\limits_{i=1}^{\left\lfloor\frac{n}{2}\right\rfloor}\binom{n}{2i}a(n-i)(-1)^{i-1}, \\
a(1) = 1
...
- 4,948
1
vote
0
answers
133
views
Sequence that sums up to A000153
Let $a(n)$ be A329369 (i.e, number of permutations of ${1,2,...,m}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \leqslant k ...
- 4,948
2
votes
2
answers
242
views
Negated Fibonacci and the floor function
Let $F_n$ be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1, \\
F_{-n} = (-1)^{n-1}F_n
$$
I conjecture that
$$
F_{-n} = \left\lfloor\frac{n+1}{2}\right\rfloor ...
- 4,948
2
votes
0
answers
28
views
On doubling or addition formulas for the sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$
We are interested which integer sequences are efficiently computable
possibly over finite rings.
Define the integer sequence $a(n)=(b_1 n +b_2)a(n-1)+(b_3 n + b_4)a(n-2)$
with initial terms $a(0),a(1)$...
- 25.4k
5
votes
0
answers
307
views
On $s$-additive sequences
For a non-negative integer $s$, a strictly increasing sequence of positive integers $\{a_n\}$ is called $s$-additive if for $n>2s$, $a_n$ is the least integer exceeding $a_{n-1}$ which has ...
- 791
1
vote
1
answer
173
views
Some ideas about parking functions and integer partitions
We know that a integer partition of $\lambda=(\lambda_1, ..., \lambda_m)$ of $n$ satisfying $\lambda_1\geq \cdots \geq \lambda_m$ and $\sum_{i=1}^m\lambda_i=n$. Let $\mathcal{P}(n)$ be the set of ...
- 11
6
votes
0
answers
245
views
Searching for a proof of the pattern and identification of integer coefficients for the A329369
Please see the update given below. Everything you need to know from the old version of the question are the functions $a(n), \ell(n), s(n), t(n), r(n)$.
Let $a(n)$ be A329369 (i.e, number of ...
- 4,948
2
votes
1
answer
261
views
Small solutions of $x^2-a^3 y^2=\pm 1$
We are interested in small integer solutions to the Pell equation:
$$x^2-a^3 y^2=\pm 1 \qquad (1)$$
Where in $\pm 1$ you can chose either sign.
$(x^2,a^3 y^2)$ are consecutive powerful numbers.
$abc$ ...
- 25.4k
1
vote
0
answers
115
views
Representing A329369 using A358612
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\dotsc,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
- 4,948
26
votes
1
answer
7k
views
Elegant recursion for A301897
Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here
$$a(n)=\frac{1}{n+1}\binom{2n}{...
- 4,948
3
votes
0
answers
128
views
Fast and simple algorithm for the A329369
Let $a(n)$ be A329369 (i.e., number of permutations of $\{1,2,\cdots,m\}$ with excedance set constructed by taking $m-i$ ($0 < i < m$) if $b(i-1) = 1$ where $b(k)b(k-1)\cdots b(1)b(0)$ ($0 \...
- 4,948
2
votes
2
answers
197
views
On the primality of $j(n)=\varphi(p_n+1-n)+1$ when $j(n) \equiv 19 \pmod {100}$
Related to Power of primes.
Let $p_n$ denote n-th prime and $\varphi$ the totient function.
For natural $n$, define $j(n)=\varphi(p_n+1-n)+1$.
For $n$ up to $10^9$ if $j(n) \equiv 19 \pmod {100}$
then ...
- 25.4k
2
votes
1
answer
131
views
Sequence that sums up to A224071
Let $a(n)$ be A224071 (i.e., number of Schroeder paths of semilength $n$ in which there are no $(2,0)$-steps at level $1$). Here
$$
a(n) = \frac{1}{2(n+1)}\sum\limits_{k=0}^{n}(k+1)((-1)^{\left\...
- 4,948
2
votes
0
answers
157
views
Conjecture: $x^4+1$ is never Wieferich prime
Related to this question and Alexander Kalmynin's answer.
For natural $n$ define $J(n)=(2^{n-1}-1) \bmod n^2$
and if $n$ is power of two define $J(2^n)=1$ (this is artificial, just to
avoid triviality ...
- 25.4k
2
votes
1
answer
236
views
An integer sequence related to Pascal’s triangle
We need someone expert in binomial coefficients (subject 11B65) to recognize the integer sequence generated by an iterative formula we have encountered while working on a project about Pascal’s ...
- 125
8
votes
4
answers
520
views
"Upside-down unimodal" sequences in combinatorics
Recall a sequence $a_0,\ldots,a_n$ of positive integers is unimodal if $a_0 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $0 \leq m \leq n$. Unimodal integer sequences are abundant in ...
- 24.2k
0
votes
0
answers
190
views
On a A057985 without recursion
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0\to01, 1\to12, 2\to0$).
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\...
- 4,948
2
votes
1
answer
120
views
Recursion for the Chebyshev transform of $m^n$
Let
$$
R(n, q, m) = R(n-1, q+1, m) + \sum\limits_{j=0}^{q} (-1)^{q-j}R(n-1, j, m), \\
R(0, q, m) = (m-1)^q
$$
I conjecture that $R(n, 0, m)$ is a Chebyshev transform of $m^n$.
Examples of Chebyshev ...
- 4,948
1
vote
1
answer
77
views
Sequence derived from transform of a given vector (with Fibonacci as partial sums)
Let F_n be A000045 (i.e., Fibonacci numbers). Here
$$
F_n = F_{n-1} + F_{n-2}, \\
F_0 = 0, F_1 = 1
$$
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). ...
- 4,948
5
votes
1
answer
200
views
Does every integer appear in the modular sum sequence?
$\newcommand{\N}{\mathbb{N}}$Let $\N$ denote the set of non-negative integers. We inductively define a sequence $a:\N\to\N$ by:
$a(0) = 0, a(1) = 1$ and
$a(n) = \big(\sum_{k=0}^{n-1}a(k)\big)\text{ ...
- 48.9k