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Questions tagged [integer-sequences]

For questions related to integer sequences (i.e. an ordered list of integers).

An integer sequence may be specified explicitly by giving a formula for its $n$th term, or implicitly by giving a relationship between its terms. For example, the sequence $0, 1, 1, 2, 3, 5, 8, 13,\dots$ (the Fibonacci sequence) is formed by starting with $0$ and $1$ and then adding any two consecutive terms to obtain the next one: an implicit description. The sequence $0, 3, 8, 15, \dots$ is formed according to the formula $n^2 − 1$ for the $n$th term: an explicit definition.

Alternatively, an integer sequence may be defined by a property which members of the sequence possess and other integers do not possess. For example, we can determine whether a given integer is a perfect number, even though we do not have a formula for the $n$th perfect number.

An integer sequence is a computable sequence if there exists an algorithm which, given $n$, calculates $a_n$, for all $n > 0$. The set of computable integer sequences is countable. The set of all integer sequences is uncountable (with cardinality equal to that of the continuum), and so not all integer sequences are computable.

Although some integer sequences have definitions, there is no systematic way to define what it means for an integer sequence to be definable in the universe or in any absolute (model independent) sense.

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If you create a power tower of all the digits in a number, calculate that out, and repeat this process, will you always reach a single digit number?

It's helpful to understand what I'm talking about here by going through some examples. Pick an arbitrary number, for this example we'll choose $37$. Creating a power tower of its digits would look like $3^7$, which is equal to 2187. Then we create…
Nico Zaczkowski
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A pattern for OEIS sequence A245869.

OEIS sequence A245869 is titled "T(n,k)=Number of length n+2 0..k arrays with some pair in every consecutive three terms totalling exactly k". Thus, for example, $T(2,3) = 100$ because there are exactly $100$ arrays consisting of $2+2=4$ numbers…
Robert Israel
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Which distances can I run on my treadmill?

This is a real-world question, prompted by some unusual features on my treadmill and which I thus think about while running. In a sub-menu on my treadmill, I can select a distance that I want to run. If my unit setting is "miles", the distances…
A. Rex
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Can the sequence $\{\lfloor \alpha n \rfloor\}$ be divided into two parts with equal sums, for all $\alpha \in \mathbb{R}$?

Define the sequence $a_n = \lfloor \alpha n \rfloor$ for a real number $\alpha$. Is there any pair of natural numbers $k, l$ satisfying the following condition?: $$\sum_{n=1}^k a_n = \sum_{n=k+1}^l a_n$$ In other words, can this sequence be divided…
dodicta
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Does this sequence involving multiplication and primorials ever return to $1$?

For clarification: as used throughout this question, the $n$th primorial is "the product of the first $n$ primes" (counting $2$ as the first prime, not the zeroth, such that the primorial of $0$ is $1$.). Define this integer sequence as follows:…
Mathemagician314
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On three integer points in the plane and a strange sequence 0, 1, 4, 5, 9, 11, 16, 25, 29, 35, 41, 55, 60, 64, 65, 80, 89, 101, 109, 125, 171, 191...

Problem 1. In the plane, three distinct fleas with integer coordinates are given. At every tick of time, two fleas sitting at the points $P$ and $Q$ can jump to two vacant points $R$ and $S$, such that $PQRS$ is a square (in that order of…
Giedrius Alkauskas
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Integer series related to the functional differential equation $f'(x) = f(f(x))$

There was an interesting discussion on the functional differential equation $$f'(x) = f(f(x)) \tag{1a}$$ The essence of which was to find a solution of $(1a)$ by Taylor expansion of $f$ about a point $x=a$, i.e. $$f(x,a) = \sum_{k\ge 0}…
Dr. Wolfgang Hintze
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Longest geometric progression of primes

There are arbitrarily long arithmetic progressions of primes e.g. $5, 11, 17, 23, 29$ for a $5$-length progression, but no (infinite) arithmetic sequence of primes with common difference $d\neq 0$, as $d\in\mathbb{Z}$ is an obvious constraint and…
A.M.
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Constructible way to find two (finite) integer sequences $(x_i)$ and $(y_j)$ just from the values of the $x_i + y_j$ sums

Let $N \geq 4$ be an integer, and suppose we have two strictly increasing finite sequences of positive integers $(x_i)$ and $(y_j)$ (where $1 \leq i \leq N$ and $1 \leq j \leq N$) such that $x_1 = y_1 = 0$. Let's also define the sums $s_{i, j} = x_i…
bghost
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Prove or reject my solution to "How quickly can you type this unary string?"

I had posted an answer on the Code Golf SE yesterday. Although the answer on that site remain valid if no counterexample can be find. I'm interesting in its correctness. So I want to find a prove or counterexample to my solution. While, answer this…
tsh
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If $(3+\sqrt{8})^{2023} = a + b\sqrt{8}$ where $a$ and $b$ are integers, then $a$ is the product of two numbers which differ by 2.

For clarity, I'm using the short-hand $\alpha = 3 + \sqrt{8}$ and $\beta = 3 - \sqrt{8}$. $\boxed{\alpha + \beta = 6\text{ and }\alpha \beta = 1}$. $\text{Define } (b_n)_{n \geq 1} := \frac{\alpha^{2n-1}+\beta^{2n-1}}{2}$ (this will always give us…
Colver
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Explicit formula for tournament sequence

I am looking for an explicit formula for a sequence. The sequence is generated as follows: There is a tournament with $10$ teams. In the beginning, all teams have a 0-0 win-loss record. The teams are paired up, and when the first round concludes,…
Jackson
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Equivalence of Definitions for "Not-Increasing" Sequences

I'm studying the concept of "not-increasing" sequences and encountered two definitions: Definition 1: $$ \exists n, m \in \mathbb{N}, \text{ such that } n < m \text{ and } a_n \geq a_m $$ Definition 2: $$ \exists n \in \mathbb{N}, \text{ such that…
BRAD ZAP
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Is the function $g(n) = \displaystyle\sum_{k=1}^{\pi(n) - 1}\left(\frac{1}{n-p_{k}}\right)$ bounded?

This is related to my other question here (that is, the terms of this sum are the reciprocals of that one). Define a function $g(n)$ that takes the sum of the reciprocals of the difference between $n$ and all the primes below it. In other…
Mathemagician314
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A convergent, but large, sum of integer reciprocals

The sums of the reciprocals of monotonic integer sequences may be convergent or diverge to infinity. (For example) Wikipedia gives examples of both. So the sum of the reciprocals of the prime numbers diverges, whereas the sum of the reciprocals of…
James K
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