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.
87
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Algorithm for the sum with binomial coefficients and Bell numbers
Let $a(n)$ be A000110 (i.e., Bell or exponential numbers: number of ways to partition a set of $n$ labeled elements).
Let $b(n)$ be A355247 (i.e., expansion of exponential generating function $\exp(2(\...
- 4,644
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0
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62
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On a number of compositions of $n$ into positive triangular numbers
Let $a(n)$ be A023361 (i.e., number of compositions of $n$ into positive triangular numbers). Here
$$
a(n) = \sum\limits_{i \geqslant 1, \frac{i(i+1)}{2}\leqslant n} a(n-\frac{i(i+1)}{2}), \\
a(0) = 1....
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vote
0
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Simple algorithm for A107670
Let $T(n, k)$ be A107670 (i.e., matrix square of triangle A107667). Here we define the triangular matrix $P$ by $P(n, k) = \frac{(n+1)^{2(n-k)}}{(n-k)!}$ for $0 \leqslant k \leqslant n$ and the ...
- 4,644
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0
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77
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Intersecting algorithm for A065601
Let $a(n)$ be A065601 (i.e., number of Dyck paths of length $2n$ with exactly $1$ hill). Here
$$
a(n) = \frac{1}{2(n+1)}((3n-2)a(n-1) + 2(9n-19)a(n-2) + 4(2n-3)a(n-3)), \\
a(0) = a(2) = 0, a(1) = 1.
$$...
- 4,644
3
votes
0
answers
163
views
Elegant algorithm for A140717
Let $T(n, k)$ be A140717 (i.e., triangle read by rows: $T(n,k)$ is the number of Dyck paths $d$ of semilength $n$ such that sum of peakheights of $d$ - number of peaks of $d$ equals $k$ ($n \geqslant ...
- 4,644
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0
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On a A347205 and related row polynomials
Let $a(n)$ be A347205. Here
$$
a(2^m(2k+1)) = \sum\limits_{j=0}^{m}a(2^j k), \\
a(0) = 1.
$$
Let $\nu_2(n)$ be A007814 (i.e., number of trailing zeros in the binary expansion of $n$). Here
$$
\nu_2(2n+...
- 4,644
2
votes
1
answer
126
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Sequence that sums up to A224071
Let $a(n)$ be A224071 (i.e., number of Schroeder paths of semilength $n$ in which there are no $(2,0)$-steps at level $1$). Here
$$
a(n) = \frac{1}{2(n+1)}\sum\limits_{k=0}^{n}(k+1)((-1)^{\left\...
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votes
0
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190
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On a A057985 without recursion
Let $a(n)$ be A057985 (i.e., start with $0$ and repeatedly substitute: $0\to01, 1\to12, 2\to0$).
Let $\operatorname{wt}(n)$ be A000120 (i.e., number of ones in the binary expansion of $n$). Here
$$
\...
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0
answers
160
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Interesting conjecture by Sequence Machine
Let $a(n)$ be A344960 (i.e., position of binary complement of $n$-th word in A341258). By definition, in order to calculate $a(n)$, we need to know A341258. Below we will correspond this sequence with ...
- 4,644
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votes
0
answers
112
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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
$$
8, 16, 32, 48, 64, ...
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votes
0
answers
103
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Formula for individual term of the Proth numbers
Let $a(n)$ be A080075 i.e. Proth numbers: of the form $k2^m + 1$ for $k$ odd, $m \geqslant 1$ and $2^m > k$.
The sequence begins with
$$
3, 5, 9, 13, 17, 25, 33, 41, 49, 57, 65, 81, 97, 113, 129
$$...
- 4,644
2
votes
0
answers
72
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Possible subsequence of the A110978
Let $a(n)$ be A110978 i.e. odd integers that are nonprime, such that there exist two factors of each number that when multiplied together in binary base, do not ever require the use of a "carry&...
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votes
2
answers
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Prime gaps within which every "small" prime appears as a factor: Are there only finitely many? Is this the last one?
For a bounded range of positive integers $n,n+1,\ldots,m,$ call a prime number "small" if it does not exceed $\sqrt m,$ so that if one is trying to factor all of these numbers into primes, ...
- 12.4k
3
votes
1
answer
248
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Prime gaps that are "relatively" bigger than all later prime gaps: Is this in OEIS?
This OEIS entry is about
Primes (lower end) with record gaps to the next consecutive prime: primes p(k) where p(k+1) - p(k) exceeds p(j+1) - p(j) for all j < k.
I'm wondering about a different ...
- 12.4k
3
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1
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973
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On the OEIS sequence A327265
The OEIS sequence https://oeis.org/A327265 starts:
$$1, 2, 5, 11, 19, 31, 51, 89, 123, 151, 179, 181, 180, 365, 634, 657, 656, 655.$$
$\mathrm{A327265}(n)$ is the smallest $k$ such that $\mathrm{...
user178594
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0
answers
99
views
Subsequence such that $c(a(n))=2^n$
Let $a(n)$ be A060831, i.e., $\sum\limits_{k=1}^{n}\operatorname{number of odd divisors of} k$.
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$b(n,k)=2b(n,k-1)-2^{k-1}, b(n,0)=n$$
Let $c(n)$ ...
- 4,644
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votes
1
answer
103
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Non-Wieferich primes with Euler quotient modulo $p$ two and alternating harmonic numbers
Let $b(n)$ denote the Euler quotient modulo $n$.
In OEIS we have A128465 Numbers k such that k divides the numerator of alternating Harmonic number H'((k+1)/2)
For $n>1$ we have $b(A128465(n))=2$.
...
- 25.2k
6
votes
1
answer
278
views
Which $n$ have $\lvert\{2^n-2^k -1\}\cap {\mathrm{PRIMES}}\rvert=m$?
Consider numbers of the form $2^n - 2^k - 1$ with $k < n$ as considered in OEIS sequence A208083. As for A208083 I investigated how many of these numbers are prime, but turned the question around: ...
- 9,670
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2
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387
views
1
vote
0
answers
190
views
Closed form for partial sums of A103318
Let $a(n)$ be A103318, number of solutions $i$ in range $[0,n-1]$ to $i \equiv 0 \pmod {2^{n-i}}$: the sequence begins with
$$1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 1, 2, 2, 2, 1, 2, 2, 3, 2$$
Also let's ...
- 4,644
1
vote
0
answers
90
views
Recurrence for the viabin numbers of the self-conjugate integer partitions
Let $a(n)$ be A290254, the viabin numbers of the self-conjugate integer partitions, also defined as $\left\lbrace 0 \right\rbrace$ union fixed points of A059894, self-inverse permutation defined as ...
- 4,644
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votes
1
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172
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Binary recurrence from general recurrence
We have general recurrence for A243499 (which is product of parts of integer partitions as enumerated in the table A125106)
$$a(n)=(1+b(n))a(t(n)), a(0)=1$$
where $b(n)$ is A023416 (which is number of ...
- 4,644
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votes
0
answers
196
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David Applegate conjecture at OEIS sequence A237424 [closed]
The OEIS sequence is the sequence of the numbers of the form $$(10^a+10^b+1)/3$$
were $a$ and $b$ are nonnegative integers
Here is the link for the sequence https://oeis.org/A237424
This sequence has ...
5
votes
1
answer
340
views
Why does this "factorial sequence" appear in the OEIS?
For a reciprocal of a polynomial, $f = \frac{1}{p}$, we (presumably) may construct a sequence $(c_n)_{n=0}^\infty$ such that for all $N\ge 0$
$$f(k)k! = \sum_{n=0}^{N-1} c_n(k-n)! + O((k-N)!). $$
I ...
- 3,499
1
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0
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191
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Generalized Thomas Ordowski conjecture at OEIS sequence A002326
OEIS is the online encyclopedia of integer sequences, Here is the link to the sequence $A002326$: https://oeis.org/A002326
For $n\geq 0$, the $n$th term in the sequence is defined as: $a(n)$ equals ...
2
votes
0
answers
325
views
Why can one compute the sum of divisors of $n$ without factoring $n$?
Question links to paper
which states:
$$
\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 ...
- 25.2k
2
votes
1
answer
498
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 ...
- 61
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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 ...
- 1,952
8
votes
1
answer
422
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 ...
- 10.4k
4
votes
2
answers
286
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
...
- 3,443
14
votes
1
answer
872
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 ...
- 26.4k
5
votes
0
answers
110
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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))...
- 155
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1
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202
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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,$$
where $b_0 = 1.$ Can someone ...
- 3,443
13
votes
2
answers
661
views
A reformulation of Erdős conjecture on arithmetic progressions
Erdős conjecture on arithmetic progressions states that if $S$ is a set of positive integers such that $c(S):=\sum_{n \in S} \frac{1}{n} = \infty$ (large set), then $ \forall \ell \ge 3$ the set $S$ ...
- 26.4k
3
votes
0
answers
263
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Guises of the refined Eulerian numbers, generated by tangent vectors (OEIS A145271)
The Eulerian numbers (OEIS A008292, not to be confused with the Euler numbers) pop up in numerous scenarios in combinatorics and advanced analysis, one as the components of the h-vectors of the ...
- 10.4k
9
votes
0
answers
222
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On the first sequence without collinear triple
Let $u_n$ be the sequence lexicographically first among the sequences of nonnegative integers with graphs without collinear three points (as for $a_n=n^2$ or $b_n=2^n$). It is a variation of that one.
...
- 26.4k
56
votes
0
answers
3k
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On the first sequence without triple in arithmetic progression
In this Numberphile video (from 3:36 to 7:41), Neil Sloane explains an amazing sequence:
It is the lexicographically first among the sequences of positive integers without triple in arithmetic ...
- 26.4k
3
votes
3
answers
628
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Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation
The entry OEIS A139605 (also related OEIS A145271) has a matrix computation for the partition polynomials that represent the expansions of iterated derivatives, or vectors in differential geometry,
$...
- 10.4k
9
votes
2
answers
674
views
Сlosed formula for $(g\partial)^n$
The objective is to obtain a closed formula for:
$$
\boxed{A(n)=\big(g(z)\,\partial_z\big)^n,\qquad n=1,2,\dots}
$$
where $g(z)$ is smooth in $z$ and $\partial_z$ is a derivative with respect to $z$.
...
- 277
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votes
1
answer
2k
views
Are there infinitely many insipid numbers?
A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ ...
- 26.4k
1
vote
1
answer
422
views
The sporadic numbers
Let call $n$ a sporadic number if the set of groups $G \neq A_n,S_n$ having a core-free maximal subgroup of index $n$ is non-empty and contains only sporadic simple groups.
By GAP, the set of all the ...
- 26.4k
8
votes
4
answers
343
views
Simple-looking sequences $A$ and $B$ defined by a complementary equation
Define $A=(a_n)$ and $B=(b_n)$ by $b_0=1$ and
$$a_n=b_n+b_{2n}$$
for $n \geq 0$, where $A$ and $B$ are increasing and every positive integer occurs exactly once in $A$ or $B$. Can someone prove ...
- 3,443
7
votes
1
answer
170
views
Number of numbers in $n$th difference sequence
Suppose that $r$ is an irrational number with fractional part between $1/3$ and $2/3$. Let $D_n$ be the number of distinct $n$th differences of the sequence $(\lfloor{kr}\rfloor)$. It appears that
$...
- 3,443
12
votes
1
answer
632
views
Integrals of power towers
Let's assume $x\in[0,1]$, and restrict all functions of $x$ that we consider to this domain. Consider a sequence $\mathcal S_n$ of sets of functions, where $n^{\text{th}}$ element is the set of all ...
- 6,809
33
votes
2
answers
844
views
A sequence potentially consisting of only integers
I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences.
Consider the sequence defined by
$$b_n = \frac{(...
- 7,875
9
votes
1
answer
208
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 ...
- 528
5
votes
0
answers
157
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 ...
- 155
4
votes
0
answers
210
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 ...
- 10.4k
6
votes
1
answer
293
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 ...
- 155
29
votes
4
answers
2k
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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 ...
- 18.6k