# Questions tagged [integer-sequences]

For questions about sequences of integers. References are often made to the online resource oeis.org.

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### Difference sequences of sets of integers

In this paper, the conception of the difference sequence and $\infty$-difference length of a subset of groups is introduced. As an important case, subsets of the additive group of integers are ...
1 vote
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### A definition related to pseudoprimes and the Dedekind psi function

In this post we consider that $\psi(k)$ denotes the Dedekind psi function. Wikipedia has an artcle dedicated to this arithmetic function Dedekind psi function defined for a positive integers $m>1$ ...
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### On gaps in a sequence of integers

Given a fixed $p \in \{3,4,5,\ldots\}$, we define the strictly increasing sequence $\{a_k\}_{k\in \mathbb N}$ as follows. We set $a_{p,1}=1$ and for each $k>1$, we set $a_{p,k}$ to be the least ...
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### Integer sequences with unique $k$-subsets sum

let the $\binom{\mathfrak{M}}{k}$ be a shorthand notation for chosing $k$ elements of set $\mathfrak{M}$ of positive integers and let $\left|\binom{\mathfrak{M}}{k}\right|$ denote the sum of the ...
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### Tangent numbers, secant numbers and permanent of matrices

Inspired by Question 402572, I consider the permanent of matrices $$f(n)=\mathrm{per}(A)=\mathrm{per}\left[\operatorname{sgn} \left(\sin\pi\frac{j+2k}{n+1} \right)\right]_{1\le j,k\le n},$$ where $n$ ...
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### Where is the source of the formula $\sum_{j=0}^\infty \bigl(j+\frac{1}{2}\bigr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}$ for an integer sequence?

The infinite series representation \begin{equation} \frac1\pi\sum_{j=0}^\infty \biggl(j+\frac{1}{2}\biggr)^{n-1}\frac{2^{j+1/2}}{\binom{2j+1}{j+1/2}}, \quad n\ge0 \end{equation} for the positive ...
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### Possible small mistake in Bilu-Hanrot-Voutier paper on primitive divisors of Lehmer sequences (?)

I think that I might have spotted I small mistake (a missing $5$-defective Lehmer pair) in the classification of terms of Lehmer sequences without primitive divisors given in: 1 Bilu, Hanrot, and ...
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### New experiments involving Ramanujan primes: Benford's law

I know that in the literature there are interesting articles involving the sequence of Ramanujan primes, I refer the Ramanujan Prime from the online encyclopedia Wolfram MathWorld. This week I ...
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### Bounds for $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$

This is related to problem in graph theory. OEIS defines A033485 as $a(1)=1$ and $a(n)=a(n-1)+a(\lfloor n/2 \rfloor)$. Q1 what are upper bounds and asymptotics for $a(n)$, can we get $\exp(o(n))$? ...
1 vote
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### Avoiding equality of partial sums of two different aperiodic sequences

Consider two distinct sequences of positive integers, $a_{n}|_{n=1}^{\infty}$, and $b_{n}|_{n=1}^{\infty}$ such that for either sequence no period exists. The elements of both sequences are drawn from ...
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### Euler quotients modulo $n$

For odd integer $n$, define the Euler quotient modulo $n$ to be $a(n)$: $$a(n)=\frac{(2^{\phi(n)}-1) \bmod n^2}{n}=\frac{2^{\phi(n)}-1}{n} \bmod n$$ $a(n)=0$ for OEIS sequence Wieferich numbers ...
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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 ...
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### Is this Laurent phenomenon explained by invariance/periodicity?

In Chapter 4 of his Tracking the Automatic Ant, David Gale discusses three families of recursively defined sequences of numbers, all due to Dana Scott and inspired by the Somos sequences: Sequence 1. ...
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### 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 ...
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: $a(n)$ ...