Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
2
votes
0
answers
32
views
joint rank sequences
An algebraic question I have been working on led me to a sequence that appears in OEIS as A186355: "adjusted joint rank sequence of $(f(i))$ and $(g(j))$ with $f(i)$ before $g(j)$ when $f(i)=g(j)$...
- 16.9k
0
votes
0
answers
63
views
Pairs of permutations such that $p(n)<2^k$ iff $n<2^k$
Let $p(n)$ be an arbitrary permutation of natural numbers such that $p(n)<2^k$ iff $n<2^k$.
Let $q(n)$ be an inverse permutation of $p(n)$.
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
...
- 4,948
5
votes
0
answers
133
views
Formula and smallest solution for the A260711
Let $a(n)$ be A260711 without initial $0$ (i.e., numbers of the form $x^2 - y^2$ with $x > y$ where $x$ and $y$ are odd, $x + y$ is a power of $2$).
The sequence begins with
$$
8, 16, 32, 48, 64, ...
- 4,948
1
vote
0
answers
68
views
On a numbers $k$ with specific $2$-adic valuation
Let $a(n)$ be A002326 (i.e., multiplicative order of $2 \operatorname{mod} 2n+1$).
Let $b(n)$ be A179382 (i.e., the smallest period of pseudo-arithmetic progression with initial term $1$ and ...
- 4,948
0
votes
0
answers
107
views
Formula for individual term of the Proth numbers
Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$.
The sequence begins with
$$
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129
$$...
- 4,948
2
votes
0
answers
72
views
Possible subsequence of the A110978
Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
- 4,948
2
votes
1
answer
236
views
An integer sequence related to Pascal’s triangle
We need someone expert in binomial coefficients (subject 11B65) to recognize the integer sequence generated by an iterative formula we have encountered while working on a project about Pascal’s ...
- 125
5
votes
1
answer
200
views
Does every integer appear in the modular sum sequence?
$\newcommand{\N}{\mathbb{N}}$Let $\N$ denote the set of non-negative integers. We inductively define a sequence $a:\N\to\N$ by:
$a(0) = 0, a(1) = 1$ and
$a(n) = \big(\sum_{k=0}^{n-1}a(k)\big)\text{ ...
- 48.9k
1
vote
0
answers
125
views
On a Fibonacci and binary
Let F(n) be A000045 i.e. Fibonacci numbers. Here
$$
F(n) = F(n-1) + F(n-2), \\
F(0) = 0, F(1) = 1
$$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
Let
$$
T(n, k) = \left\lfloor\frac{n}{2^k}\...
- 4,948
1
vote
1
answer
147
views
Strongly regular binary sequences
Let $\mathbb{N} = \{0,1,2,\ldots\}$ denote the set of non-negative integers. If $n\in\mathbb{N}$ we let $[n] = \{0,\ldots,n-1\}$. For $A
\subseteq \mathbb{N}$ we let $$\mu^+(A) = \lim\sup_{n\to\infty}\...
- 48.9k
0
votes
0
answers
62
views
Linear recurrences in coefficients of powers of quotients of polynomial rings
It is known that linear recurrences with constant coefficients
can be computed via powers in $\mathbb{Z}[x]/f(x)$.
We believe that this generalizes to quotients of multivariate polynomial
rings.
Let $...
- 25.4k
2
votes
2
answers
197
views
On the primality of $j(n)=\varphi(p_n+1-n)+1$ when $j(n) \equiv 19 \pmod {100}$
Related to Power of primes.
Let $p_n$ denote n-th prime and $\varphi$ the totient function.
For natural $n$, define $j(n)=\varphi(p_n+1-n)+1$.
For $n$ up to $10^9$ if $j(n) \equiv 19 \pmod {100}$
then ...
- 25.4k
2
votes
1
answer
281
views
Curious sequences of polynomials
Given an integer $k\geq 2$, and $k+1$ invertible initial
values $s_0,s_1,\ldots,s_k$ in some commutative ring $\mathcal A$
we set
$$s_{n+1}=\frac{\sum_{j=1}^ks_{n+1-j}^2+q \sum_{j=1}^{k-1}s_{n+1-j}s_{...
- 17.9k
2
votes
1
answer
113
views
Natural density of thickly syndetic set
A syndetic set $S$
is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural ...
2
votes
1
answer
268
views
A problem similar to the $3x+1$-problem [closed]
Let $n$ be a fixed positive integer. Define the function $f_n(x)$ as follows:
$$f_n(x)=\left\{\begin{aligned}&2x-1,\quad x\leq n;\\&2(x-n),\quad x> n.\end{aligned}\right.$$
and for $l\in\...
- 111
2
votes
1
answer
177
views
An upper bound on coefficients of some integer sequences
Given $\lambda>0$ let $B=B(\lambda)$ be the smallest integer
such that there exist infinite integer sequences
having values in $\lbrace 1,2,\ldots,B-1,B\rbrace$ and satisfying
the following ...
- 17.9k
2
votes
2
answers
210
views
An identity for the ratio of two partial Bell polynomials
Let $B_{\ell,m}(x_1,x_2,\dotsc,x_{\ell-m+1})$ denote the Bell polynomials of the second kind (or say, partial Bell polynomials, (exponential) partial Bell partition polynomials). I knew that
the ...
- 1,101
1
vote
0
answers
109
views
Can the ideas of convex optimization be used to prove a bound?
If we define $\lambda(n)=\lfloor \log_2(n) \rfloor$ and $v(n)$ as the binary digit sum of positive integer $n$ we can make a toy example of what I think is the most important conjecture in addition ...
- 97
3
votes
0
answers
120
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
- 4,948
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
13
votes
1
answer
700
views
When is $\mathrm{gcd}(k,p^k-1)=1$ true?
Let $p$ be a prime. Is there a classification of the numbers $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$?
For the ...
- 63.1k
4
votes
0
answers
156
views
The smallest sequence without differences among Fibonacci numbers
Given a subset $\mathcal S\subset \mathbb N\setminus\{0\}$
of (strictly) positive integers, we can consider subsets
$A$ of $\mathbb N$ (or $\mathbb Z$) with no differences in
$\mathcal S$.
Examples: ...
- 17.9k
0
votes
1
answer
122
views
Permutation of the natural numbers from operation related to binary expansion of $n$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
Let $T(n,k)$ be a $(k+1)$-th bit from the right side in the binary expansion of $n$. Here
$$
T(n, k) = \left\lfloor\frac{n}{2^k}\right\rfloor \...
- 4,948
2
votes
0
answers
199
views
Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
- 4,948
3
votes
0
answers
69
views
Sequence that sum up to A343685
Let $a(n)$ be A343685 i.e.
$$
a(n)=2na(n-1)+\sum\limits_{j=0}^{n-1}\binom{n}{j}(n-j-1)!a(j), \\
a(0)=1
$$
Here the exponential generating function $A(x)$ satisfy
$$
A(x)=\frac{1}{1-2x+\log(1-x)}
$$
...
- 4,948
3
votes
1
answer
233
views
Min problem on integers
Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log_2 n)/1000$ integers $1=q_1\le q_2\le \cdots q_{s-1}\le q_s=n$. Prove that
$$\min_\ell\left(\sum_{i=1}^\ell q_i\right)\left(\sum_{i=...
1
vote
0
answers
111
views
Recursion for the Bessel polynomial $y_n(x)$
Let $a(n)$ be A001515 i.e. the Bessel polynomial $y_n(x)$ evaluated at $x=1$. Here
$$
a(n) = (2n-1)a(n-1) + a(n-2), \\
a(0) = 1, a(1) = 2
$$
The closed form is
$$
a(n)=\sum\limits_{k=0}^{n}\binom{n+k}{...
- 4,948
1
vote
0
answers
94
views
Combinatorial interpretation for the more general case of $R(n,0)$
Let $f(n), g(n,m), h(n)$ be an arbitrary functions which equal to the non-negative integers.
Let
$$
R(n,q) = \sum\limits_{j=0}^{f(q)}g(q,j)R(n-1,j),\\
R(0,q) = h(q)
$$
In the comment to the one of ...
- 4,948
0
votes
1
answer
140
views
Series reversion using something like continued fraction
Let $f(n)$ be an arbitrary function such that $f(n)\in\mathbb{Z}$.
Let
$$
F(x)=\sum\limits_{m\geqslant 0}f(m)x^m
$$
Define the operator $\operatorname{SR}$, which is associated with the series ...
- 4,948
26
votes
1
answer
7k
views
Elegant recursion for A301897
Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here
$$a(n)=\frac{1}{n+1}\binom{2n}{...
- 4,948
4
votes
0
answers
121
views
$f(n) = \frac{n^2 + n + 4}{2}$, $g(f(n)) = f(g(n))$ such that $g(n)$ is an integer
Let $n$ be a strict positive integer and let's define an integer sequence $f(n)$ :
$$f(n) = \frac{n^2 + n + 4}{2}$$
so
$$
\begin{split}
f (\Bbb N)& \triangleq {3,5,8,12,17,23,30,38,47,\ldots}\\
f(...
- 763
2
votes
0
answers
126
views
Recurrence for A004208
Let $a(n)$ be A004208. Here
$$a(n)=n\prod\limits_{j=1}^{n}(2j-1)-\sum\limits_{i=1}^{n-1}a(i)\prod\limits_{j=1}^{n-i}(2j-1)$$
I conjecture that
$$a(n)=R(n-1,0)$$
where
$$R(n,q)=2(q+2)R(n-1,q+1)+\sum\...
- 4,948
2
votes
0
answers
100
views
Sequence of numbers related to line-segment intersections
Question:
what is known about the sequence $\mathbb{X}\subset \mathbb{N}_0$ such that for each $k\in \mathbb{X}$ there exists a set of $n$ points in general position in the Euclidean plane such that ...
- 13.2k
4
votes
0
answers
306
views
How to explain this number-theoretic seeming “almost coincidence”?
For natural numbers $n\geq2$, let $d(n)$ be the number of divisors of $n$, and let
\begin{equation}
g(n)=n\sum_i r_i(p_i-1)
\end{equation}
where $n=\prod_i p_i^{r_i}$ is the factorisation of $n$ as a ...
- 141
-2
votes
1
answer
168
views
Two-variable continuous function which results in an integer if and only if arguments are integer
I am looking for functions $f(x,y)$, real arguments, continuous,
with the following properties:
$f(m,n) = r$, where $r$ is integer $> 0$ if and only if $m,n$ are integers $> 0$.
$f(m,n) \le f(...
- 3
0
votes
0
answers
60
views
Existence of integer sequence under simultaneous constraints
Does there exist a function $f:\Bbb N\to\Bbb N$ such that \begin{align}a_{n+1}&=f(a_n)\\a_{f(n)+1}&=a_n\end{align} implies $\{a_n\}_{n\ge0}$ is a non-constant, positive integer sequence? ...
- 1,459
8
votes
0
answers
88
views
Generalization of Lucas sequences to order 3 (and above)
For fixed integer parameters $(P,Q)$, Lucas sequences represent a pair of complimentary integer sequences satisfying the same recurrence with the characteristic polynomial $f(x):=x^2 - Px + Q$. The ...
- 34.4k
3
votes
1
answer
140
views
Sequences that sum up to Dowling numbers
Let $a(n,k)$ be the sequence of $k$-Dowling numbers (for more information see A007405 and its CROSSREFS section) with e.g.f.
$$\operatorname{exp}\left(x + \frac{\operatorname{exp}(kx) - 1}{k}\right)$$
...
- 4,948
2
votes
0
answers
105
views
Sequences that sum up to the many sequences in the OEIS
Let
$$a(n,m,k)=\frac{1}{n}\sum\limits_{j=0}^{n}[n+kj\geqslant 0]\binom{n}{j}\binom{n+kj}{j-1}(m-1)^{j-1}$$
Here square brackets denote Iverson brackets.
There are many sequences in the OEIS that are ...
- 4,948
6
votes
1
answer
268
views
Sequence that sums up to the number of permutations avoiding the pattern $1-23-4$
Let $a(n)$ be A113227, i.e., the number of permutations on $[n]\equiv \{1, \ldots, n\}$ avoiding the pattern $1-23-4$.
The sequence begins with
$$1, 1, 2, 6, 23, 105, 549, 3207, 20577, 143239, 1071704,...
- 4,948
2
votes
0
answers
70
views
Integer coefficients such that $T(n,k)=R(n,k)-R(n,k-1)$
Let $a(n)$ be A000085, i.e., the number of self-inverse permutations on $n$ letters, also known as involutions; number of standard Young tableaux with $n$ cells. Here
$$a(n) = a(n-1) + (n-1)a(n-2), a(...
- 4,948
1
vote
0
answers
57
views
Recurrence for the number of permutations with a given excedance set
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
$$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...
- 4,948
1
vote
0
answers
134
views
Recurrence for the A284005
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
$$f(n)=n-2^{\ell(n)}$$
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
$$\operatorname{wt}(2n+1)=\operatorname{wt}(n)+1, \...
- 4,948
2
votes
0
answers
76
views
Uniqueness of the permutation
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
- 4,948
1
vote
0
answers
182
views
Ask for a proof of an inequality involving the Bernoulli numbers
Let $B_k$ be the Bernoulli numbers and let
\begin{equation}
T_k=\frac{2^{2k}}{(2k)!}|B_{2k}|, \quad k\ge1.
\end{equation}
Prove the inequality
\begin{equation*}
\frac{\frac{1}{k+2}\sum_{j=0}^{k+1}\...
- 1,101
1
vote
0
answers
109
views
Existence of binary permutations with a given property
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$f(n)=n-2^{\ell(n)}$$
Let $a(n)$ be a permutation of the nonnegative integers such that $a(0)=0$, $a(n)=n$ if $n$ is a power of $2$ and ...
- 4,948
2
votes
1
answer
172
views
Permutation and its binary analog
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
- 4,948
1
vote
0
answers
81
views
Infiniteness of the pairs of sequences with a given conditions
Let
$$\varphi=\frac{1+\sqrt{5}}{2}$$
Let
$$a_1(n)=\left\lfloor n\varphi \right\rfloor, a_2(n)=n+a_1(n)$$
Let $\operatorname{tr}(n)$ be A007814, i.e., the number of trailing zeros in the binary ...
- 4,948
0
votes
0
answers
61
views
Stolarsky array and Stolarsky representation
Let $T(n,k)$ be A035506, i.e., Stolarsky array read by antidiagonals. Here we consider that $T(n,k)=0$ for $n<1, k<1$.
Let $a(n)$ be A200714, i.e., Stolarsky representation interpreted as binary ...
- 4,948
1
vote
1
answer
114
views
Coefficients of number of the same terms which are arising from iterations based on binary expansion of $n$
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$T(n,k)=\left\lfloor\frac{n}{2^k}\right\rfloor\operatorname{mod}2$$
Here $T(n,k)$ is the $(k+1)$-th bit from the right side in the binary ...
- 4,948