Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
399 questions
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Iterations of $2^{n-1}+5$: the strong law of small numbers, or something bigger?
I've discovered what I believe is a quite remarkable sequence (A318970), defined by
$$n_1 = 3,\qquad n_{k+1} = 2^{n_k-1}+5\quad(k\geq 1).$$
Here are the first four terms with their prime ...
- 34.4k
49
votes
4
answers
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Strange (or stupid) arithmetic derivation
Let us consider the following operation on positive integers: $$n=\prod_{i=1}^{k}p_i^{\alpha_i} \qquad f(n):= \prod_{i=1}^{k}\alpha_ip_i^{\alpha_i-1}$$ (Is it true that if we apply this operation to ...
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2
answers
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Can we find lattice polyhedra with faces of area 1,2,3,...?
I asked this question two months ago on MSE, where it earned the rare
Tumbleweed badge for garnering zero votes, zero answers, and 25 views over 61 days.
Perhaps justifiably so! Here I repeat it with ...
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1
answer
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Mod sequences that seem to become constant; and the number 316
Define a "mod sequence" of nonnegative integers
based on one start parameter $s$, its first term,
as follows.
$A(s)=(a_1,a_2,\ldots,a_n,\ldots)$
with $a_1 = s$
and
$$ a_n = \left(\sum_{k=1}^{n-1} a_k \...
- 151k
35
votes
8
answers
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Examples of integer sequences coincidences
For the time being, the OEIS website contains almost $300000$ sequences. Each of these sequences is the mark of a specific mathematical concept. Sometimes two (or more) distinct concepts have the ...
35
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0
answers
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Is there any positive integer sequence $c_{n+1}=\frac{c_n(c_n+n+d)}n$?
In a recent answer Max Alekseyev provided two recurrences of the form mentioned in the title which stay integer for a long time. However, they eventually fail.
QUESTION Is there any (added: ...
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34
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1
answer
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A remarkable almost-identity
OEIS sequence A210247 gives the signs of $\text{li}(-n,-1/3) = \sum_{k=1}^\infty (-1)^k k^n/3^k$, also the signs of the Maclaurin coefficients of $4/(3 + \exp(4x))$.
Mikhail Kurkov noticed that it ...
- 54.2k
33
votes
2
answers
856
views
A sequence potentially consisting of only integers
I will first ask the question which can be stated very simply. Afterwards I will explain some motivation and give references to related sequences.
Consider the sequence defined by
$$b_n = \frac{(...
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33
votes
0
answers
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The easily bored sequence
If we want to compare the repetitiveness of two finite words, it looks reasonable, first of all, to consider more repetitive the word repeating more times one of its factors, and secondarily to ...
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32
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0
answers
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A question related to the Hofstadter–Conway \$10000 sequence
The Hofstadter–Conway \$10000 sequence is defined by the nested recurrence relation $$c(n) = c(c(n-1)) + c(n-c(n-1))$$ with $c(1) = c(2) = 1$. This sequence is A004001 and it is well-known that this ...
- 701
27
votes
1
answer
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Why do the adjoint representations of three exceptional groups have the same first eight moments?
For a representation of a compact Lie group, the $n$th moment of the trace of that representation against the Haar measure is the dimension of the invariant subspace of the $n$th tensor power. The ...
- 149k
26
votes
1
answer
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Elegant recursion for A301897
Let $a(n)$ be A301897, i.e., number of permutations $b$ of length $n$ that satisfy the Diaconis-Graham inequality $I_n(b) + EX_n(b) \leqslant D_n(b)$ with equality. Here
$$a(n)=\frac{1}{n+1}\binom{2n}{...
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26
votes
1
answer
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A surprising conjecture about twin primes
Just for fun, I began to play with numbers of two distinct ciphers. I noticed that most of the cases if you consider the numbers $AB$ and $BA$ (written in base $10$), these have few common divisors: ...
- 363
26
votes
3
answers
907
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What is the smallest size of a shape in which all fixed $n$-polyominos can fit?
Let $n$ be an integer and consider all fixed $n$-polyominos, i.e., without rotation or reflection. I am interested in finding a shape in which all polyominos can embed. (It is OK if multiple ...
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24
votes
1
answer
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Parity of the multiplicative order of 2 modulo p
Let $\operatorname{ord}_p(2)$ be the order of 2 in the multiplicative group modulo $p$. Let $A$ be the subset of primes $p$ where $\operatorname{ord}_p(2)$ is odd, and let $B$ be the subset of primes $...
- 429
24
votes
0
answers
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Is A276175 integer-only?
The terms of the sequence A276123, defined by $a_0=a_1=a_2=1$ and $$a_n=\dfrac{(a_{n-1}+1)(a_{n-2}+1)}{a_{n-3}}\;,$$ are all integers (it's easy to prove that for all $n\geq2$, $a_n=\frac{9-3(-1)^n}{2}...
- 655
23
votes
3
answers
2k
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Zeroes of the random Fibonacci sequence
Let $X_n$ be the "random Fibonacci sequence," defined as follows:
$X_0 = 0, X_1 = 1$;
$X_n = \pm X_{n-1} \pm X_{n-2}$, where the signs are chosen by independent 50/50 coinflips.
It is known ...
- 19.2k
23
votes
5
answers
1k
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Sequences with integral means
Let $S(n)$ be the sequence whose first element is $n$, and from then onward,
the next element is the smallest natural number ${\ge}1$ that ensures that the
mean of all the numbers in the sequence is ...
- 151k
22
votes
1
answer
2k
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Reference request: a tale of two mathematicians
I've heard tell the following anecdote involving Pierre Gabriel and Jacques Tit at least twice in a lapse of four years or so:
When P. Gabriel presented the theorem in a conference [sometime around ...
- 8,842
20
votes
13
answers
7k
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Longest coinciding pair of integer sequences known
There are arbitrarily many pairs of integer sequences (of arbitrary origins) that coincide upto an $N$ but differ for an $n > N$. I assume, the coincidence will be considered accidentally then by ...
- 9,720
20
votes
2
answers
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A possibly surprising appearance of $\sqrt{2}.$
Define $A=(a_n)$ and $B=(b_n)$ as follows: $a_0=1$, $a_1=2$, $b_0=3$, $b_1=4$, and $$a_n=a_1b_{n-1}-a_0b_{n-2} + 2n$$ for $n \geq 2$, where $A$ and $B$ are increasing and every positive integer occurs ...
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19
votes
2
answers
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A finite alternating sum
We have stumbled upon the following finite alternating sum, which we have trouble analyzing. The sum is:
$$
S_n = \sum_{j=0}^n \frac{ (-1)^j e^{-j} }{j!} (n-j)^j
$$
We have observed numerically that ...
- 193
19
votes
1
answer
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Is every sequence that looks like an AP really an AP?
Caveat: I am not at all a number theorist, and I randomly came up with the following question while I was hiking. But I already asked two serious number theorists, and since they did not know the ...
- 11.9k
19
votes
1
answer
1k
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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 ...
- 25.4k
19
votes
2
answers
581
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Sequences with 3 letters
For a positive integer $n$ I would like to construct long sequences consisting of 0, 1 and 2's such that for any two subsequences consisting of $n$ consecutive elements the number of 0's , 1's or 2'...
- 2,286
18
votes
8
answers
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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 ...
- 1,625
18
votes
2
answers
992
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A conjecture harmonic numbers
I will outlay a few observations applying to the harmonic numbers that may be interesting to prove (if it hasn't already been proven).
From the Online Encyclopedia of Positive Integers we have:
$a(n)$ ...
- 181
18
votes
1
answer
607
views
Order of Conway's "look and say" recurrence
Let $L_n$ be the length of the $n$th term of Conway's "look and say"
sequence (https://oeis.org/A005341). The generating function $F(x)=
\sum_{n\geq 0}L_nx^n$ is a rational function, say $P(x)/Q(x)$ ...
- 50.8k
17
votes
2
answers
3k
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Some unpublished notes of Hofstadter
I'm looking for some unpublished notes called "Eta Lore," which are apparently related to a talk Douglas Hofstadter first gave at the Stanford Math Club in 1963. I know these notes exist because they'...
16
votes
2
answers
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Does this sequence ever end?
This may help: A080670 A195265
Define $f(n)$ as this:
Take a number $n$, and split it into its prime composition using $^$ and $×$. Now remove all $^$ and $×$, you get a new number, this is $f(n)$ (...
- 467
16
votes
2
answers
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are these polynomials or rationals functions?
Let $x$ be a variable. Define the following family of sequences (reminiscent of Lucas polynomials) according to the rule: $P_0(x):=0, P_1(x):=1$ and for $n\geq2$ by
$$P_n(x)=xP_{n-1}(x)-P_{n-2}(x).$$
...
- 43.2k
15
votes
1
answer
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Arithmetic progressions in stopping time of Collatz sequences
Inspired by the question here, we did a few more simulations of numbers of some specific forms and noticed a pattern.
We consider the original $3n+1$ transform where we divide by $2$ if it's even and ...
- 259
15
votes
1
answer
475
views
Determinant of a matrix filled with elements of the Thue–Morse sequence
Let $n$ be a positive integer. Suppose we fill a square matrix $n\times n$ row-by-row with the first $n^2$ elements of the Thue–Morse sequence (with indexes from $0$ to $n^2-1$). Let $\mathcal D_n$ be ...
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15
votes
1
answer
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Conjecture on signed sum of integer fractions x/y from 1..N?
Here is a generalization of an integer challenge that was asked on Yahoo!Answers in 2009, I believe it could be original, defies induction and has exponential-complexity. Not aware of any theory that ...
15
votes
0
answers
523
views
Convergence of rivers of numbers
$\DeclareMathOperator{\river}{river}\DeclareMathOperator{\leadingsum}{ls}\DeclareMathOperator{\digitsum}{ds}\newcommand{\qed}{\square}
$A 1999 British Informatics Olympiad question asks about ...
- 159
15
votes
0
answers
487
views
Word complexity of primes mod 4
For an infinite binary word $w$, the word complexity $f_w(n)$ is defined as the number of different subwords of length $n$. The asymptotic behavior of this function is an important parameter of the ...
- 17.1k
14
votes
4
answers
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Integrality of a sequence formed by sums
Consider the following sequence defined as a sum
$$a_n=\sum_{k=0}^{n-1}\frac{3^{3n-3k-1}\,(7k+8)\,(3k+1)!}{2^{2n-2k}\,k!\,(2k+3)!}.$$
QUESTION. For $n\geq1$, is the sequence of rational numbers $a_n$ ...
- 43.2k
14
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5
answers
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Is the sequence $a_n=c a_{n-1} - a_{n-2}$ always composite for $n > 5$?
Numerical evidence suggests the following.
For $c \in \mathbb{N}, c > 2$ define the sequence $a_n$ by
$a_0=0,a_1=1, \; a_n=c a_{n-1} - a_{n-2}$
For $ 5 < n < 500, \; 2 < c < 100$ there ...
- 25.4k
14
votes
1
answer
697
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Are the asymptotics of A003238 known?
Sequence A003238 of the OEIS counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins
1, 1, 2, 3, 5, 6, 10, 11, 16, ...
and it is ...
- 1,305
14
votes
2
answers
2k
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sequences with a fractal dimension
This is inspired by the self-similarity of the celebrated Golay-Rudin-Shapiro sequence, more exactly, of its alternating partial sums. (This latter one is oeis 020990). The pictures show the 550 first ...
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14
votes
1
answer
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Put as many points as possible in an equilateral triangle of side 1 with their minimal distance greater than 1/n
It is known by the pigeon-hole principle that:
If we select $5$ points within an equilateral triangle with side $1$, there must be at least two whose distance apart is less than or equal to $1/2$.
...
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14
votes
1
answer
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A Collatz-like question about permutations
An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting.
Consider all permutations $\pi$ on the natural numbers such that ...
- 37.7k
14
votes
1
answer
835
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Special configurations on a circle from a homological algebra problem
Here is the short version of the combinatorial problem:
Given a positive integer $n \geq 2$. Draw a circle with $2n$ points indexed by the numbers from $\mathbb{Z}/ 2n \mathbb{Z}$. We colour the ...
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13
votes
2
answers
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Positive integers written as $\binom{w}2+\binom{x}4+\binom{y}6+\binom{z}8$ with $w,x,y,z\in\{2,3,\ldots\}$
Let $\mathbb N=\{0,1,2,\ldots\}$. Recall that the triangular numbers are those natural numbers
$$T_x=\frac {x(x+1)}2\quad \text{with}\ x\in\mathbb N.$$
As $T_x=\binom{x+1}2$, Gauss' triangular number ...
- 15.6k
13
votes
1
answer
700
views
When is $\mathrm{gcd}(k,p^k-1)=1$ true?
Let $p$ be a prime. Is there a classification of the numbers $k \geq 1$ such that $\gcd(k,p^k-1)=1$? If not, can we at least produce an explicit infinite subset? What is known about these $k$?
For the ...
- 63.1k
13
votes
0
answers
718
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Is "OEIS A001935 Number of partitions with no even part repeated" efficiently computable $\mod 4$?
Is A001935 Number of partitions with no even part repeated efficiently computable $\mod 4$?
I am interested because of this relation with sum of divisors of $8n+1$.
$\sigma(8n+1) \equiv A001935(n) \...
- 25.4k
12
votes
4
answers
1k
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Six consecutive positive integers with certain shape
Are there 6 consecutive positive integers, where each of them is a square or the product of a prime and a square ?
If they exist, one of those six integers A will be the product of 2 and a square of ...
- 321
12
votes
2
answers
1k
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Asymptotic behavior of the sequence $u_n = u_{n-1}^2-n$
I am currently interested in the following sequence:
$$\begin{cases}u_0 & = & \alpha\\u_n & = & u_{n-1}^2-n\end{cases}$$ where $\alpha > C \approx 1.75793275... $ with $C$ being the ...
- 123
12
votes
1
answer
427
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Subwords of the infinite Fibonacci word
Let $W = 01001010010010 \ldots$ be the infinite Fibonacci word, A003849 in the OEIS. Let $B(m)$ be the set of $m+1$ subwords of $W$ that have length $m$, and for each such subword $u$, let $p(u)$ be ...
- 479
12
votes
1
answer
634
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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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