Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [oeis]

The acronym OEIS stands for the On-Line Encyclopedia of Integer Sequences, a well-known database of sequences of integers. It can be used for questions where this database is (or might be) relevant, mainly questions about particular sequences of integers. This tag is typically used in combination with other tags to make the scope of the question more precise; common examples of such tags include the top-level tags co.combinatorics and nt.number-theory.

8
votes
1answer
138 views

Reference for Kakutani result on power sum bases of symmetric functions

Numerical semigroups are additive submonoids $A$ of the natural numbers such that the greatest common divisor of all elements of $A$ is 1. The complement of a numerical semigroup in $\mathbb{N}$ is ...
3
votes
0answers
110 views

Dirichlet eta function and Stirling Permutations

The Stirling permutations of order $k$ is a permutation of the multiset $1, 1, 2, 2, ..., k, k$. The Dirichlet $\eta$-function is a function closely related to the Riemann $\zeta$-function. According ...
3
votes
0answers
88 views

Conjecture on tilting modules for an Auslander algebra

On page 13 of "Tilting modules for the Auslander algebra of $K(x)/x^n$" the author, Geuenich, suggests that the number ($p_{n,i}$) of isomorphism classes of modules, occurring as the $i$-th summand of ...
4
votes
1answer
92 views

Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?

In OEIS A003436, it is written that the number of inequivalent labeled Hamilton Cycles of an n-dimesnional Octahedron is the same as the number of Perfect Matchings in a the complement of the Cycle ...
24
votes
3answers
1k views

Advanced software for OEIS?

Is there (if not, why?) a software where I can input a sequence of integers, like into the OEIS, and then it makes some simple transformations on it to check whether the sequence can be obtained from ...
0
votes
1answer
207 views

Ordinary Generating Function for OEIS A056296?

The sequence OEIS A056296 can be obtained using $ a(n)={1\over n}\sum_{d\backslash n}\varphi(d)\begin{cases} {n/d+2\brace3}-{n/d+1\brace3}, & \text{$6\backslash d$;} \\ {n/d+2\brace3}-3{n/d+...
29
votes
7answers
3k views

Examples of integer sequences coincidences

For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
2
votes
1answer
113 views

Counting particular Dyck paths

The OEIS entry for Pascal’s triangle contains the following intriguing remark: $C(n,k)$ = the number of Dyck paths of semilength $n$, with $k$ "u"'s in odd numbered positions and $k$ returns to the ...
10
votes
1answer
712 views

Ramanujan's Lost Notebook page 1 first equation and OEIS sequence A260195

In the 1988 Narosa edition of Ramanujan's The Lost Notebook and Other Unpublished Papers, on the first line of page 1 is the following: $$ \Big(1+\frac1a\Big) \Bigg\{\frac{1}{(1-aq)(1-q/a)}+\frac{q(1+...
1
vote
2answers
164 views

Linear Extension of the $n\times n$ lattice

I am looking at a particular integer sequence, the number of $n\times n$ Young Tableaus (see OEIS). In the comments at OEIS, Mitch Harris stated that the same sequence also defines the Number of ...
1
vote
1answer
209 views

OEIS Sequence A002846 and properties of matrix inverses

Sequence A002846 in https://oeis.org/A002846 (OEIS) gives, for each positive integer $n$, the number $a(n)$ of ways of transforming a set of $n$ indistinguishable objects into $n$ singletons via a ...
0
votes
0answers
80 views

Conjectured congruence $A048852(n-1) \equiv p_n - p_{n-1} \pmod{4}$

Let $p_n$ denote the n-th prime. OEIS A048852 is shortly defined as "Difference between b^2 (in c^2=a^2+b^2) and product of successive prime pairs". Numerical evidence for $3 \le n \le 52$, all of ...
8
votes
4answers
948 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 ...
2
votes
0answers
117 views

Link of a power series by the Bernoullis for a Riccati equation to zonotopes?

On pg. 85 of The Rise and Development of Theory of Series up to the Early 1820s by Ferraro is a series soln. of $$ d^2z/z = -x^2dx^2 $$ related to the reputed first appearance of a Riccati-type eqn.,...
2
votes
0answers
142 views

Claim in OEIS will give some results about Legendre's and Brocard's conjectures

Claim in OEIS will give non-trivial results about Legendre's and Brocard's conjecture. The claim is very likely to be true, but I am not sure it is currently provable. Brocard's conjecture states ...
0
votes
1answer
92 views

Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $D(N^2)$ is the deficiency of $N^2$?

Any results on $\gcd(N^2, D(N^2))$ where $N^2$ is deficient and $$D(N^2)=2N^2 - \sigma(N^2)$$ is the deficiency of $N^2$? I checked OEIS sequence A033879 and have so far been able to get hold of ...
7
votes
0answers
194 views

Increasing derivatives of recursively defined polynomials

Consider recursively defined polynomials $f_0(x)=x$ and $f_{n+1}(x)=f_n(x)−f_n'(x) x (1−x)$. These polynomials have some special properties, for example $f_n(0)=0$, $f_n(1)=1$, and all $n+1$ roots of ...
2
votes
1answer
196 views

Relating face polytopes of permutohedra to integer partitions

The OEIS entries A019538, A049019, and A133314, relate a refinement of the face polynomials of the permutohedra (A049019) to partition polynomials (A133314) defined by multiplicative inversion of an ...
3
votes
0answers
199 views

First to note/document the relation between permutohedra and multiplicative inversion

The relation between the refined face numbers of the permutohedra and the formal series expansion of the reciprocal of a function (exponential generating function, formal Taylor series) is given in ...
6
votes
1answer
481 views

Frankl's conjecture and Oeis sequence A188163

For every natural number $c \geq 2$, let $f(c)$ denote the least natural number $f$ with the following property : every union-closed family of sets with at least $f$ members has $c$ members whose ...
4
votes
1answer
265 views

Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m. I don't see why this ...
5
votes
1answer
237 views

Up to $2000$, $A145722(n-1) \equiv \sigma(4n-3) \pmod{5}$

A145722 is Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions. Using the pari program and offset 0, up to $2000$...
7
votes
0answers
188 views

Has anyone seen these binary trees (Catalan-type related to the Gegenbauer polynomials and Motzkin paths)?

The OEIS entry A121448 enumerates binary trees with $n$ edges and $k$ vertices with outdegree 1. Has anyone seen these trees? The o.g.f. for this entry, $G(x,t)$, is essentially a discriminant ...
8
votes
2answers
496 views

Asymptotics of product of Euler's totient function (A001088)?

Conjecture: \begin{align} \lim_{n\to \infty } \, \frac{\left(\prod _{k=1}^n \phi (k)\right){}^{1/n}}{n}\sim 0.2059\text{...} \end{align} The numerical result from 100000 terms is: My questions are: ...
7
votes
1answer
738 views

Another formula for Bell numbers

Here is an observation (thanks to OEIS): $$\sum_{i=0}^\infty \frac{i^k}{i!}= B_k e,$$ where $B_k$ is the $k$-th Bell number. I might be having reading comprehension issues, but I don't see this ...
3
votes
1answer
107 views

Upper bound for OEIS A076689 “Smallest k such that k*p#+1 is prime”?

OEIS A076689 Is defined as smallest integer $a(n)=k$ such that $k n\#+1$ is prime, where $n\#$ is primorial, the product of the first $n$ primes. Lower bound appears $1$, the primorial primes. ...
18
votes
10answers
1k views

Combinatorial Databases

At one point, I remember being excited by seeing the website Encyclopedia of Combinatorial Structures as an extension of Sloane's Online Integer Sequence Database site. Unfortunately, the site (ECS) ...
22
votes
1answer
1k views

Wrong asymptotics of OEIS A000607 (number of partitions of an integer in prime parts)?

Sequence A000607 in the Online Encyclopedia of Integer Sequences is the number of partitions of $n$ into prime parts. For example, there are $5$ partitions of $10$ into prime parts: $10 = 2 + 2 + 2 + ...
11
votes
4answers
2k views

Ordinary Generating Function for Bell Numbers

In the OEIS entry for Bell numbers, there appears a generating function $$\sum_{k=0}^\infty B_k t^k = \sum_{r=0}^\infty \prod_{i=1}^r \frac{t}{1-it}$$ However, I could not locate any proof of ...
3
votes
2answers
435 views

Are all counterexamples of OEIS A226181 both Poulet numbers and Proth numbers?

OEIS A226181: 3, 5, 7, 11, 13, 17, 19, 23, 29, 37, 41, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 113, 131, 137, 139, 149, 163, ... Primes $p$ ...
41
votes
2answers
2k views

Numbers that are generic w.r.t. exponentiation

This is a follow-up to my old question Number of distinct values taken by $x\hat{\phantom{\hat{}}}x\hat{\phantom{\hat{}}}\dots\hat{\phantom{\hat{}}}x$ with parentheses inserted in all possible ways. ...
23
votes
1answer
837 views

Busy Beaver modulo 2

There is well-known Rado's "Busy Beaver" sequence — the maximal number of marks which a halting Turing machine with n states, 2 symbols (blank, mark) can produce onto an initially blank two-way ...
0
votes
1answer
541 views

Number of labeled regular graphs on n vertices

What is known about the number of labeled regular graphs on n vertices? The sequence does not appear to be in the OEIS.
9
votes
0answers
623 views

Is there a proof that OEIS-A002387 is $[ e^{n-\gamma} ]$?

Based on the comments on OEIS-A002387: $a_{n}$ = 1, 2, 4, 11, 31, 83, 227, 616,... it is likely, that the sequence $a_{n}$ coincides with $[ e^{n-\gamma} ]$ , where $\gamma$ is the Euler-Mascheroni ...
11
votes
2answers
2k views

How to cite a sequence from The On-Line Encyclopedia of Integer Sequences (OEIS)?

In my paper I want to provide a reference for a sequence (in this case - A001970) from The On-Line Encyclopedia of Integer Sequences (OEIS). However, I couldn't find an official bib entry for it (...
14
votes
0answers
854 views

Is OEIS A007018 really a subsequence of squarefree numbers?

A comment in A007018 a(n) = a(n-1)^2 + a(n-1), a(0)=1 claims Subsequence of squarefree numbers (A005117). - Reinhard Zumkeller, Nov 15 2004 Is that really so? As far as I know, it is an open ...
16
votes
1answer
1k views

Number of distinct values taken by $\alpha$ ^ $\alpha$ ^ $\dots$ ^ $\alpha$ with parentheses inserted in all possible ways, $\alpha\in\mathbf{Ord}$

Let $\alpha\in\mathbf{Ord}$ and $n\in\mathbb{N}^+$. Let $F_\alpha(n)$ be the number of distinct values taken by ordinal exponentiation $\underbrace{\alpha \hat{\phantom{\hat{}}} \alpha \hat{\phantom{\...
0
votes
1answer
201 views

Understanding a sequence generation formula of the A064532

I'm trying to understand the formula presented for the sequence A064532 from the OEIS, looks like a recurrence relation with complex numbers: $a(10i+j) = a(i) + a(j), etc.$ Sorry if its a simple ...
6
votes
1answer
469 views

Complexity of computing expansion of a newform level 18 weight 3 and character [3] - OEIS A116418

I am not familiar with newforms, so this may not make any sense. OEIS sequence A116418 is Expansion of a newform level 18 weight 3 and character [3] Numerical evidence suggest that up to $10^5$ $$ \...
18
votes
1answer
1k views

Number of distinct values taken by x^x^…^x with parentheses inserted in all possible ways

For what positive x's the number of distinct values taken by x^x^...^x with parentheses inserted in all possible ways is not represented by the sequence A000081? Is it exactly the set of positive ...
11
votes
1answer
834 views

Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ (Number of partitions with no even part repeated )

Up to $10^6$: $\sigma(8n+1) \mod 4 = OEIS A001935(n) \mod 4$ A001935 Number of partitions with no even part repeated Is this true in general? It would mean relation between restricted partitions ...
21
votes
2answers
1k views

Least number of non-zero coefficients to describe a degree n polynomial

I'd be grateful for a good reference on this, it feels like a classic subject yet I couldn't find much about it. Polynomials in one variable of the form $x^n+a_{n-1}x^{n-1}+\dots +a_1 x+a_0$ can be ...