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.

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Recurrence for the viabin numbers of the self-conjugate integer partitions

Let $a(n)$ be A290254, the viabin numbers of the self-conjugate integer partitions, also defined as $\left\lbrace 0 \right\rbrace$ union fixed points of A059894, self-inverse permutation defined as ...
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1 answer
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Binary recurrence from general recurrence

We have general recurrence for A243499 (which is product of parts of integer partitions as enumerated in the table A125106) $$a(n)=(1+b(n))a(t(n)), a(0)=1$$ where $b(n)$ is A023416 (which is number of ...
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2 votes
0 answers
182 views

David Applegate conjecture at OEIS sequence A237424 [closed]

The OEIS sequence is the sequence of the numbers of the form $$(10^a+10^b+1)/3$$ were $a$ and $b$ are nonnegative integers Here is the link for the sequence https://oeis.org/A237424 This sequence has ...
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3 votes
1 answer
271 views

Why does this "factorial sequence" appear in the OEIS?

For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$ $$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$ I ...
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1 vote
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Generalized Thomas Ordowski conjecture at OEIS sequence A002326

OEIS is the online encyclopedia of integer sequences, Here is the link to the sequence $A002326$: https://oeis.org/A002326 For $n\geq 0$, the $n$th term in the sequence is defined as: $a(n)$ equals ...
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1 vote
0 answers
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Why can one compute the sum of divisors of $n$ without factoring $n$?

Question links to paper which states: $$ \sigma(n)= \frac{6}{n^2(n-1)}\sum_{k=1}^{n-1}(3n^2-10k^2)\sigma(k)\sigma(n-k) \qquad (1) $$ where $\sigma(n)$ is the sum of divisors of $n$. Another similar ...
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1 vote
1 answer
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What OEIS sequence is this?

I've come up with an idea of an integer sequence. It can be formulated (perhaps a bit loosely) as follows: For n points N(n) is the number of configurations where each point either lies on some ...
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7 votes
0 answers
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My research paper involves computing additional terms of an existing OEIS sequence. Should I first amend the sequence or publish the results?

In the course of my research I computed terms of an existing OEIS sequence that are currently unknown. Having prepared my paper for publication, I am now faced with a (small) dilemma: Do I first ...
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  • 1,841
8 votes
1 answer
391 views

Conjecture by Ekedahl on Weyl groups and Abelian varieties

A conjecture was made on p.14 in "Cycle Classes of the E-O Stratification on the Moduli of Abelian Varieties" by Torsten Ekedahl (late, excellent contributor to MO) and Gerard Van Der Geer concerning ...
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3 votes
2 answers
183 views

Positions in the Wythoff array

Suppose that $x$ and $y$ are positive integers. How can the position of $x+y$ in the Wythoff array (A035513) be predicted from the positions of $x$ and $y$? Background. The Wythoff array begins with ...
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12 votes
1 answer
447 views

On the iterated automorphism groups of the cyclic groups

Let $C_n$ be the cyclic group of order $n$. Its automorphism group $Aut(C_n)$ is a group of order $\varphi(n)$ isomorphic to $(\mathbb{Z}/n\mathbb{Z})^{\times}$ the multiplicative group of integer ...
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5 votes
0 answers
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Hypergraphs with only disjoint perfect matchings

Let $H(n,r)$ be the set of $r$-uniform hypergraph with $n$ vertices that have only disjoint perfect matchings (i.e. every hyperedge only appears in at most one of the perfect matchings). Let $m(h(n,r))...
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4 votes
1 answer
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Difference of two integer sequences: all zeros and ones?

Suppose that $c$ is a nonnegative integer and $A_c = (a_n)$ and $B_c = (b_n)$ are strictly increasing complementary sequences satisfying $$a_n = b_{2n} + b_{4n} + c,$$ where $b_0 = 1.$ Can someone ...
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12 votes
2 answers
527 views

A reformulation of Erdős conjecture on arithmetic progressions

Erdős conjecture on arithmetic progressions states that if $S$ is a set of positive integers such that $c(S):=\sum_{n \in S} \frac{1}{n} = \infty$ (large set), then $ \forall \ell \ge 3$ the set $S$ ...
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3 votes
0 answers
243 views

Guises of the refined Eulerian numbers, generated by tangent vectors (OEIS A145271)

The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the ...
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6 votes
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Guises of the Noncrossing Partitions (NCPs)

From "Noncrossing partitions in surprising locations" by Jon McCammond: "Certain mathematical structures make a habit of reoccuring in the most diverse list of settings. Some obvious examples ...
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9 votes
0 answers
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On the first sequence without collinear triple

Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one. ...
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50 votes
0 answers
2k views

On the first sequence without triple in arithmetic progression

In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence: It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
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3 votes
3 answers
464 views

Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation

The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry, $...
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9 votes
2 answers
592 views

Сlosed formula for $(g\partial)^n$

The objective is to obtain a closed formula for: $$ \boxed{A(n)=\big(g(z)\,\partial_z\big)^n,\qquad n=1,2,\dots} $$ where $g(z)$ is smooth in $z$ and $\partial_z$ is a derivative with respect to $z$. ...
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6 votes
1 answer
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Are there infinitely many insipid numbers?

A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...
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1 vote
1 answer
386 views

The sporadic numbers

Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups. By GAP, the set of all the ...
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8 votes
4 answers
328 views

Simple-looking sequences $A$ and $B$ defined by a complementary equation

Define $A=(a_n)$ and $B=(b_n)$ by $b_0=1$ and $$a_n=b_n+b_{2n}$$ for $n \geq 0$, where $A$ and $B$ are increasing and every positive integer occurs exactly once in $A$ or $B$. Can someone prove ...
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7 votes
1 answer
158 views

Number of numbers in $n$th difference sequence

Suppose that $r$ is an irrational number with fractional part between $1/3$ and $2/3$. Let $D_n$ be the number of distinct $n$th differences of the sequence $(\lfloor{kr}\rfloor)$. It appears that $...
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11 votes
1 answer
505 views

Integrals of power towers

Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
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26 votes
0 answers
516 views

A sequence potentially consisting of only integers

I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences. Consider the sequence defined by $$b_n = \frac{(...
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9 votes
1 answer
175 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 ...
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5 votes
0 answers
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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 ...
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4 votes
0 answers
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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 ...
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6 votes
1 answer
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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 ...
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27 votes
3 answers
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 ...
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0 votes
1 answer
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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+...
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33 votes
7 answers
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
1 answer
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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 ...
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12 votes
1 answer
865 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+...
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2 votes
2 answers
317 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 ...
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1 vote
1 answer
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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 ...
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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 ...
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9 votes
4 answers
1k 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 ...
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2 votes
0 answers
144 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.,...
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2 votes
0 answers
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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 ...
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0 votes
1 answer
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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 ...
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7 votes
0 answers
206 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 ...
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2 votes
1 answer
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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 ...
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3 votes
0 answers
211 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 ...
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  • 7,920
7 votes
1 answer
521 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 ...
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  • 1,205
4 votes
1 answer
279 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 ...
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5 votes
1 answer
250 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 $2000$...
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7 votes
0 answers
213 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 ...
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  • 7,920
9 votes
2 answers
663 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: ...
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