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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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, ...
Notamathematician's user avatar
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0 answers
92 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 $$ 3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129 $$...
Notamathematician's user avatar
2 votes
0 answers
67 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&...
Notamathematician's user avatar
6 votes
2 answers
1k views

Prime gaps within which every "small" prime appears as a factor: Are there only finitely many? Is this the last one?

For a bounded range of positive integers $n,n+1,\ldots,m,$ call a prime number "small" if it does not exceed $\sqrt m,$ so that if one is trying to factor all of these numbers into primes, ...
Michael Hardy's user avatar
3 votes
1 answer
234 views

Prime gaps that are "relatively" bigger than all later prime gaps: Is this in OEIS?

This OEIS entry is about Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k. I'm wondering about a different ...
Michael Hardy's user avatar
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 > ...
Notamathematician's user avatar
3 votes
1 answer
940 views

On the OEIS sequence A327265

The OEIS sequence https://oeis.org/A327265 starts: $$1, 2, 5, 11, 19, 31, 51, 89, 123, 151, 179, 181, 180, 365, 634, 657, 656, 655.$$ $\mathrm{A327265}(n)$ is the smallest $k$ such that $\mathrm{...
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1 vote
0 answers
97 views

Subsequence such that $c(a(n))=2^n$

Let $a(n)$ be A060831, i.e., $\sum\limits_{k=1}^{n}\operatorname{number of odd divisors of} k$. Let $$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$ Let $$b(n,k)=2b(n,k-1)-2^{k-1}, b(n,0)=n$$ Let $c(n)$ ...
Notamathematician's user avatar
0 votes
1 answer
95 views

Non-Wieferich primes with Euler quotient modulo $p$ two and alternating harmonic numbers

Let $b(n)$ denote the Euler quotient modulo $n$. In OEIS we have A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2) For $n>1$ we have $b(A128465(n))=2$. ...
joro's user avatar
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6 votes
1 answer
275 views

Which $n$ have $\lvert\{2^n-2^k -1\}\cap {\mathrm{PRIMES}}\rvert=m$?

Consider numbers of the form $2^n - 2^k - 1$ with $k < n$ as considered in OEIS sequence A208083. As for A208083 I investigated how many of these numbers are prime, but turned the question around: ...
Hans-Peter Stricker's user avatar
1 vote
2 answers
371 views

Sum of reciprocals of A086005

Does the sum of reciprocals of terms of A086005 converge?
Daniel Sebald's user avatar
1 vote
0 answers
188 views

Closed form for partial sums of A103318

Let $a(n)$ be A103318, number of solutions $i$ in range $[0,n-1]$ to $i \equiv 0 \pmod {2^{n-i}}$: the sequence begins with $$1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 2$$ Also let's ...
Notamathematician's user avatar
1 vote
0 answers
89 views

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 ...
Notamathematician's user avatar
0 votes
1 answer
172 views

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 ...
Notamathematician's user avatar
2 votes
0 answers
193 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 ...
Ahmad Jamil Ahmad Masad's user avatar
4 votes
1 answer
323 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 ...
Zach Hunter's user avatar
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1 vote
0 answers
184 views

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 ...
Ahmad Jamil Ahmad Masad's user avatar
2 votes
0 answers
314 views

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 ...
joro's user avatar
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2 votes
1 answer
472 views

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 ...
A Z's user avatar
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7 votes
0 answers
195 views

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 ...
Klangen's user avatar
  • 1,943
8 votes
1 answer
412 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 ...
Tom Copeland's user avatar
  • 9,897
4 votes
2 answers
268 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 ...
Clark Kimberling's user avatar
14 votes
1 answer
733 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 ...
Sebastien Palcoux's user avatar
5 votes
0 answers
108 views

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))...
Mario Krenn's user avatar
4 votes
1 answer
196 views

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 ...
Clark Kimberling's user avatar
13 votes
2 answers
627 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 $S$ ...
Sebastien Palcoux's user avatar
3 votes
0 answers
261 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 ...
Tom Copeland's user avatar
  • 9,897
9 votes
0 answers
218 views

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. ...
Sebastien Palcoux's user avatar
53 votes
1 answer
3k 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 ...
Sebastien Palcoux's user avatar
3 votes
3 answers
612 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, $...
Tom Copeland's user avatar
  • 9,897
9 votes
2 answers
661 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 $z$. ...
Wakabaloola's user avatar
6 votes
1 answer
2k views

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 $486$ ...
Sebastien Palcoux's user avatar
1 vote
1 answer
416 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 ...
Sebastien Palcoux's user avatar
8 votes
4 answers
340 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 ...
Clark Kimberling's user avatar
7 votes
1 answer
169 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 $...
Clark Kimberling's user avatar
11 votes
1 answer
606 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 ...
Vladimir Reshetnikov's user avatar
28 votes
0 answers
559 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{(...
John Machacek's user avatar
9 votes
1 answer
204 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 ...
tghyde's user avatar
  • 508
5 votes
0 answers
151 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 ...
Mario Krenn's user avatar
4 votes
0 answers
207 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 ...
Tom Copeland's user avatar
  • 9,897
6 votes
1 answer
284 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 ...
Mario Krenn's user avatar
28 votes
4 answers
2k 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 ...
domotorp's user avatar
  • 18.3k
1 vote
1 answer
258 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+...
Robert A. Russell's user avatar
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
184 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 ...
Robin Houston's user avatar
14 votes
1 answer
911 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+...
Somos's user avatar
  • 2,464
2 votes
2 answers
446 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 ...
Skrodde's user avatar
  • 329
1 vote
1 answer
222 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 ...
Helmut's user avatar
  • 169
0 votes
0 answers
89 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 ...
joro's user avatar
  • 24.2k
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 ...
Anton's user avatar
  • 1,573