Questions tagged [integer-sequences]

For questions about sequences of integers. References are often made to the online resource oeis.org.

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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}{...

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