46
votes
Examples of integer sequences coincidences
A historical example, in the sense that the conjectural equality has been refuted: A180632 (Minimum length of a string of letters that contains every permutation of $n$ letters as sub-strings) was ...
30
votes
Examples of integer sequences coincidences
[EDITED]
The classic example is A000396: "Perfect numbers n: n is equal to the sum of the proper divisors of n"
and A000668(n)*(A000668(n)+1)/2 where A000668 are the Mersenne primes.
They are the ...
23
votes
Advanced software for OEIS?
There is a "superseeker" option at OEIS which does something like what you are asking for.
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19
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Advanced software for OEIS?
Perhaps worth mentioning here Sagemath's functionality to communicate with OEIS. (Sagemath is basically a large Python library).
Apart from searching, one can retrieve components of records, etc etc.
...
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17
votes
Accepted
Are there infinitely many insipid numbers?
Almost all $n$ are insipid. In fact, the number of non-insipid numbers at most $n$ grows like $2n/\log n$. See the paper
Cameron, Peter J.; Neumann, Peter M.; Teague, David N.
On the degrees of ...
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16
votes
Accepted
Advanced software for OEIS?
There has been some previous discussion on the OEIS mailing list about similar topics. For instance, about the sum of two sequences. (If I recall correctly, there was a university project that ...
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14
votes
Examples of integer sequences coincidences
Just another instance of the (second) Strong Law of Small Numbers:
We have A157656(n) = A059100(n-1) for all known terms (i.e., $n\leq 6$), but it's also known that A157656(29) > A059100(28). So, the ...
13
votes
How to cite a sequence from The On-Line Encyclopedia of Integer Sequences (OEIS)?
The OEIS website http://oeis.org/wiki/Works_Citing_OEIS#Referencing_the_OEIS provides this info:
If you have found the OEIS useful in your own work, and wish to reference it, the usual citation is
<...
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13
votes
Computationally challenging integer sequences
I think the question is too broad. There are a variety of such examples, and it seems that many combinatorial items and computational complexity items would fit the bill. The OEIS has a "most ...
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13
votes
Ramanujan's Lost Notebook page 1 first equation and OEIS sequence A260195
This conjecture is equivalent to the following
$$\frac{q}{(1-q)^2}\sum_{n=0}^\infty(-q)^n \frac{(q;q^2){}_n(-q^2;q^2){}_n}{(q^3;q^2){}_n^2}=\sum_{1\le r,s\le t}q^{t^2-\frac{1}{2}(r^2-r+s^2-s)},\tag{1}$...
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12
votes
Accepted
Prime gaps within which every "small" prime appears as a factor: Are there only finitely many? Is this the last one?
As noted by Will Jagy in the comments, this is closely related to the size of prime gaps: Any gap of size at least $\sqrt{m}$ has this property.
In fact, every gap with this property has size at least ...
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11
votes
Сlosed formula for $(g\partial)^n$
In OEIS A124796 I considered a similar problem of computing the coefficients of $(\partial_z\circ M_g)^n$, where $M_g$ is the operator of multiplying by $g(z)$.
It turns out that the coefficients ...
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10
votes
Examples of integer sequences coincidences
Some sequences have conjectured formulas, for example the following which I encountered recently, A227404.
$a_n$ is the total number of inversions, among all permutations on $[n]$ that is a single ...
10
votes
Computationally challenging integer sequences
Here is a sequence that is quite famous but not in OEIS I think (perhaps because too few elements are known). A hypergraph is $v$-uniform if every edge has $v$ vertices, and 2-colourable if the ...
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9
votes
Computationally challenging integer sequences
How about busy beaver numbers. I think the first 5 are known, and there are very large lower bounds for the 6th and 7th, which may already be uncomputible. The sequence definitely becomes ...
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9
votes
Accepted
Why is the number of Hamiltonian Cycles of n-octahedron equivalent to the number of Perfect Matching in specific family of Graphs?
The $n$-dimensional analogue of the octahedron is the complement of a perfect matching of its vertex set. (Every vertex is joined to every other vertex except its antipode.) If you take a Hamilton ...
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9
votes
A reformulation of Erdős conjecture on arithmetic progressions
This question is basically asking how good greedy-type constructions of sets without long arithmetic progressions can be. The answer is actually pretty terrible.
Firstly, as you note, if $f_k(n)$ is ...
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9
votes
Accepted
What OEIS sequence is this?
Your sequence is the same as the linked OEIS sequence. This is the
Number of partitions of $n$ into parts of two kinds.
In your case, the two kinds are circles for which the centre is occupied and ...
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8
votes
How to cite a sequence from The On-Line Encyclopedia of Integer Sequences (OEIS)?
For those who, like me, just want to copy and paste BibTeX for OEIS which approximates the general suggested reference style, here you go:
...
8
votes
Accepted
Reference for Kakutani result on power sum bases of symmetric functions
I think Zagier must have been thinking of the following papers of Kakeya, instead of Kakutani
Kakeya, S.: On fundamental systems of symmetric functions. I, II. Jap. J. Math.2, 69–80 (1925) ; 4, 77–...
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7
votes
Is OEIS A007018 really a subsequence of squarefree numbers?
Prime factors below $10^{10}$ of $a_n$ can be found in OEIS A007996, and I've tested that none of them divides $a_n$ when squared. Same was reported by Andersen earlier for primes below $2^{32}$.
In ...
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7
votes
A sequence potentially consisting of only integers
This is more of a comment on your sequences than an answer to your question; unfortunately I do not have enough points to post a comment yet.
The sequences $(a_n^{(k)})_{n=0}^\infty$ labelled by $k=2, ...
6
votes
Computationally challenging integer sequences
The Hales-Jewett numbers $c_{n,k}$ are defined, essentially, to be the largest possible size of a subset of $\mathbb Z_k^n$ free of $k$-term arithmetic progressions with the difference in $\{0,1\}^n$ ...
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6
votes
On the iterated automorphism groups of the cyclic groups
It seems to me that the structure of ${\rm Aut}^{m}(C_{n})$ also depends heavily on the prime factorization of $n$, and I don't really see any reason to expect the answer to Q1 to be any more ...
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6
votes
Polynomials for natural numbers and irreducible polynomials for prime numbers?
This doesn't answer the main question (Edit: I have a different answer now that does), but it addresses the later edit (and is a bit too long for a comment): we do not have $|\theta-2|>1$ for all ...
- 1,343
5
votes
Computationally challenging integer sequences
Longest non-repeating sequence of Conway's game of life states, on an $n \times n$-torus, A294241.
In order to compute the $n$th term, one needs to compute the longest path in a graph with $2^{n^2}$ ...
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5
votes
Accepted
Linear Extension of the $n\times n$ lattice
The $n \times n$ lattice just means the product poset of two chains: $[0,n] \times [0,n]=\{(i,j) | 0 \leq i,j \leq n\}$ where $(i,j) \leq (i',j')$ if and only if $i \leq i'$ and $j \leq j'$.
A linear ...
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5
votes
The sporadic numbers
I cannot give a complete answer to this question right now, but I believe that it would be possible to answer it by writing a moderate amount of computer code that made use of existing results in the ...
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5
votes
Accepted
Expansions of iterated, or nested, derivatives, or vectors--conjectured matrix computation
I have finally written up the proof in detail. It is in my note
Darij Grinberg, Commutators, matrices and an identity of Copeland, also available as arXiv:1908.09179v1.
Your result is a particular ...
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