Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
358
questions
3
votes
1
answer
355
views
Ask for a generating function or an explicit expression of a triangle of positive integers
Preliminaries
I encountered the following triangle of positive integers:
$c_{n,k}$
$n=1$
$n=2$
$n=3$
$n=4$
$n=5$
$n=6$
$n=7$
$n=8$
$k=0$
$1$
$3$
$15$
$105$
$315$
$3465$
$45045$
$45045$
$k=1$
$5$
$...
4
votes
0
answers
104
views
Do all nonnegative integers appear in A051521?
For every positive integer $n$, $\tau(n)$ is the number of divisors of $n$. If we list the ratio of each positive integer $n$ to $\tau(n)$,they form a rational sequence
1,1,3/2,4/3,5/2,3/2,…
Because $\...
11
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 ...
7
votes
1
answer
385
views
Counting hyperplane arrangements up to combinatorial equivalence, simple examples and history
Two arrangements of (affine) hyperplanes in $d$-dimensional Euclidean space are combinatorially isomorphic (or combinatorially equivalent) if they have isomorphic posets of faces.
Counting the ...
3
votes
0
answers
84
views
Increasing integral sequence of intermediate growth which is periodic modulo almost all primes
Many integral sequences are periodic modulo (almost) all primes.
However all examples I know are either evaluations of suitable polynomials on consecutive integers (trivial examples) or grow at least ...
2
votes
0
answers
29
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 $f(i)=g(j)$...
0
votes
0
answers
56
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
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
...
5
votes
0
answers
77
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
$$
8, 16, 32, 48, 64, ...
1
vote
0
answers
61
views
On a numbers $k$ with specific $2$-adic valuation
Let $a(n)$ be A002326 (i.e., multiplicative order of $2 \operatorname{mod} 2n+1$).
Let $b(n)$ be A179382 (i.e., the smallest period of pseudo-arithmetic progression with initial term $1$ and ...
8
votes
0
answers
1k
views
Is the Collatz conjecture known to be true for interesting unbounded classes of numbers?
The Collatz or the $3n+1$ conjecture is open.
Is there a specific polynomial $f(x)\in\mathbb Z[x]$ whose range is unbounded for which every integer of form $|f(m)|$ at $m\in\mathbb Z$ satisfies $3n+1$...
0
votes
0
answers
93
views
Formula for individual term of the Proth numbers
Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$.
The sequence begins with
$$
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129
$$...
2
votes
0
answers
69
views
Possible subsequence of the A110978
Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
2
votes
1
answer
224
views
An integer sequence related to Pascal’s triangle
We need someone expert in binomial coefficients (subject 11B65) to recognize the integer sequence generated by an iterative formula we have encountered while working on a project about Pascal’s ...
5
votes
1
answer
189
views
Does every integer appear in the modular sum sequence?
$\newcommand{\N}{\mathbb{N}}$Let $\N$ denote the set of non-negative integers. We inductively define a sequence $a:\N\to\N$ by:
$a(0) = 0, a(1) = 1$ and
$a(n) = \big(\sum_{k=0}^{n-1}a(k)\big)\text{ ...
1
vote
0
answers
115
views
On a Fibonacci and binary
Let F(n) be A000045 i.e. Fibonacci numbers. Here
$$
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}\...
1
vote
1
answer
130
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 $$\mu^+(A) = \lim\sup_{n\to\infty}\...
0
votes
0
answers
58
views
Linear recurrences in coefficients of powers of quotients of polynomial rings
It is known that linear recurrences with constant coefficients
can be computed via powers in $\mathbb{Z}[x]/f(x)$.
We believe that this generalizes to quotients of multivariate polynomial
rings.
Let $...
2
votes
2
answers
187
views
On the primality of $j(n)=\varphi(p_n+1-n)+1$ when $j(n) \equiv 19 \pmod {100}$
Related to Power of primes.
Let $p_n$ denote n-th prime and $\varphi$ the totient function.
For natural $n$, define $j(n)=\varphi(p_n+1-n)+1$.
For $n$ up to $10^9$ if $j(n) \equiv 19 \pmod {100}$
then ...
41
votes
2
answers
2k
views
Can we find lattice polyhedra with faces of area 1,2,3,...?
I asked this question two months ago on MSE, where it earned the rare
Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with ...
2
votes
1
answer
257
views
Curious sequences of polynomials
Given an integer $k\geq 2$, and $k+1$ invertible initial
values $s_0,s_1,\ldots,s_k$ in some commutative ring $\mathcal A$
we set
$$s_{n+1}=\frac{\sum_{j=1}^ks_{n+1-j}^2+q \sum_{j=1}^{k-1}s_{n+1-j}s_{...
1
vote
2
answers
336
views
Are there infinitely long arithmetic progressions in every increasing sequence of positive integers with bounded gaps between consecutive terms?
Suppose the largest gap is D>1 and at least two of the gaps 1,2,...,D appear infinitely many times. I think the answer is NO. But I find it difficult to formulate a necessary and sufficient ...
3
votes
0
answers
162
views
Closed form for $a(2^m(2^n-2^p-1))$
Let $q(n)$ be A007814, i.e., the number of trailing zeros in the binary representation of $n$. Here
$$q(2n+1)=0, q(2n)=q(n)+1$$
Let $a(n)$ be A329369. Here
$$a(2n+1)=a(n), a(2n)=a(n)+a(n-2^{q(n)})+a(...
2
votes
1
answer
95
views
Natural density of thickly syndetic set
A syndetic set $S$
is a subset of the natural numbers $\mathbb{N}$ or integers $\mathbb{Z}$, having the property of "bounded gaps": that the sizes of the gaps in the sequence of natural ...
1
vote
1
answer
251
views
A problem similar to the $3x+1$-problem [closed]
Let $n$ be a fixed positive integer. Define the function $f_n(x)$ as follows:
$$f_n(x)=\left\{\begin{aligned}&2x-1,\quad x\leq n;\\&2(x-n),\quad x> n.\end{aligned}\right.$$
and for $l\in\...
5
votes
0
answers
967
views
A generalization of the difference of squares identity
Let us find explicit integer functions for the coefficients of the monomial expansion of
$$
Q \left( x_1, \ldots , x_n \right) = \prod_{\left( \kappa_1, \ldots , \kappa_{n-1} \right) \in \{-1,1\}^{n-1}...
0
votes
0
answers
53
views
Stolarsky representation from Zeckendorf representation with some pairs of bits inverted
Let $a(n)$ be A200714 i.e. Stolarsky representation interpreted as binary to decimal integers.
Let $b(n)$ be A003714 i.e. Fibbinary numbers (Zeckendorf representation interpreted as binary to decimal ...
18
votes
8
answers
2k
views
Computationally challenging integer sequences
I wonder what are the examples of integer sequences, where only few elements are known and the researchers are still actively looking for the new terms. I think this discussion might be a good ...
4
votes
2
answers
503
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 ...
2
votes
2
answers
196
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 ...
20
votes
13
answers
7k
views
Longest coinciding pair of integer sequences known
There are arbitrarily many pairs of integer sequences (of arbitrary origins) that coincide upto an $N$ but differ for an $n > N$. I assume, the coincidence will be considered accidentally then by ...
2
votes
1
answer
171
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 ...
0
votes
0
answers
78
views
Partitions in A237981
Let $T(n,k)$ be A237981 i.e. array: row $n$ gives the NW partitions of n; see Comments. Here by $T(n,k)$ I mean $k$-th partition in $n$-th row.
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
...
1
vote
0
answers
106
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 ...
0
votes
0
answers
68
views
Recursions for the A111528
Let $T(n,k)$ be A111528 i.e. square table, read by antidiagonals, where the g.f. for row $n+1$ is generated by
$$
xg_{n+1}(x) = \frac{1}{n+1}\left(1+nx - \frac{1}{g_n(x)}\right), \\
g_0(x) = \sum\...
3
votes
0
answers
115
views
Sequence which is related to the binary expansion of $n$ and partition numbers
Let $p(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers).
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let $\operatorname{wt}(n)$ be A000120 i.e. number of $1$'s in ...
8
votes
4
answers
489
views
"Upside-down unimodal" sequences in combinatorics
Recall a sequence $a_0,\ldots,a_n$ of positive integers is unimodal if $a_0 \leq \cdots \leq a_m \geq \cdots \geq a_n$ for some $0 \leq m \leq n$. Unimodal integer sequences are abundant in ...
13
votes
1
answer
656
views
When is $\mathrm{gcd}(k,p^k-1)=1$ true?
Let $p$ be a prime. Is there a classification of the numbers $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$?
For the ...
1
vote
0
answers
98
views
Conjecture on numbers $k$ having only one partition into parts with same binary weight as a binary weight of $k$
Let $\operatorname{tr}(n)$ be A007814, number of trailing zeros in the binary representation of $n$.
Also, let $\operatorname{ntr}(n)$ be A086784, number of non-trailing zeros in the binary ...
2
votes
0
answers
199
views
Not a twin prime pair test using $\gcd$ only
Let $m$ be an odd positive integer such that $m=2k+1$, $k\in\mathbb{N}$.
Let $v$ be a vector of $n$ positive integers. Let $v(i)$ be the $i$-th element of the vector. Then we start with $v(i)=m(i+1)-2$...
4
votes
0
answers
136
views
The smallest sequence without differences among Fibonacci numbers
Given a subset $\mathcal S\subset \mathbb N\setminus\{0\}$
of (strictly) positive integers, we can consider subsets
$A$ of $\mathbb N$ (or $\mathbb Z$) with no differences in
$\mathcal S$.
Examples: ...
1
vote
2
answers
160
views
Weirdness in the sequence "the number of divisors for a weird number"
I thought it would be fun to give my froshling students a short programming assignment to characterize numbers as: deficient, abundant, perfect, and prime. Then I got a little carried away and started ...
0
votes
1
answer
101
views
Permutation of the natural numbers from operation related to binary expansion of $n$
Let
$$
\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 \...
0
votes
0
answers
77
views
Constructing a pair of complementary sequences with the perfect differences
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 $g(n,m)$ be A257961. Here
$$
g(n, m) = mF_{n-1} \operatorname{mod} F_n
$$
Let
$$
\varphi=\...
3
votes
1
answer
798
views
What do we know about Lucky numbers?
I'm really fascinated by lucky numbers (Wikipedia; OEIS A000959) and their prime-like characteristics.
Wolfram states: write "out all odd numbers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, .... The ...
0
votes
0
answers
62
views
Simple non-recursive formula for inverse permutation to A316385
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
Let
$$
f(n)=n+2^{\ell(n)+1}
$$
Let $a(n)$ be A316385, i.e. lexicographically earliest sequence of distinct positive terms such that for any $n > ...
1
vote
2
answers
217
views
Values of the Kronecker symbol of recursive sequences
Is there something known about the values of the Kronecker symbol $\left(\frac{a_n}{a_{n+1}}\right)$ if $a_n$ is a recursive sequence? I know something about this if $a_n$ is the Fibonacci sequence or ...
3
votes
0
answers
67
views
Sequence that sum up to A343685
Let $a(n)$ be A343685 i.e.
$$
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)}
$$
...
3
votes
1
answer
222
views
Min problem on integers
Let $n$ be any integer greater than $2^{10^6}$. Given any $s\le (\log_2 n)/1000$ integers $1=q_1\le q_2\le \cdots q_{s-1}\le q_s=n$. Prove that
$$\min_\ell\left(\sum_{i=1}^\ell q_i\right)\left(\sum_{i=...
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
vote
0
answers
104
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}{...