Questions tagged [integer-sequences]
For questions about sequences of integers. References are often made to the online resource oeis.org.
420 questions
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Conjecture: linear recurrences with constant coefficients satisfy non-linear irreducible polynomial [closed]
Working over the integers, probably any field will do.
Let $a(n)$ be linear recurrence with constant coefficients:
$a(n)=c_1 a(n-1)+c_2 a(n-2)+\cdots +c_d a(n-d)$.
Conjecture 1 There exist integers $k,...
- 25.8k
2
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Are there primitive weird numbers that are multiple of a cube of an odd prime?
A positive integer $n$ is weird if it is abundant and not semiperfect, i.e., it cannot be expressed as a sum of distinct proper divisors of $n$.
A trivial consequence of the definition of weird number ...
- 662
5
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1
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On the non-squarefree odd part of a weird number
A positive integer $n$ is weird if it is abundant and cannot be expressed as a sum of distinct proper divisors of $n$. As in the case of perfect numbers, all weird numbers currently known are even (in ...
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Number of integer solutions of $n = \operatorname{lcm}(x,y) - \gcd(x,y)$ with $1 \leqslant y < x \leqslant n$
Let
$\omega(n)$ be the prime omega function such that it counts the number of distinct prime factors of $n$.
$a(n)$ be an integer sequence such that $n$-th term is the number of integer solutions of $...
1
vote
1
answer
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Recurrence for columns of A125790
Let
$\operatorname{wt}(n)$ be A000120, i.e., the number of ones in the binary expansion of $n$. Here $$ \operatorname{wt}(2n+1) = \operatorname{wt}(n)+1, \\ \operatorname{wt}(2n) = \operatorname{wt}(...
5
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0
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Is oeis.org/A378562 unbounded?
$a(n)$ is the number of steps to reach $1$ by reversing digits of $n$ in the lowest base where reversal reduces the number, as defined in A378562.
For example, $9 = 1001_2$, so base $2$ does not ...
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1
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The existence of closed form solutions for Ta(4)
While there are known examples of numbers expressible as a sum of two positive integer cubes in four distinct ways like $6963472309248 = 2421^3 + 19083^3 = 5436^3 + 18948^3 = 10200^3 + 18072^3 = 13320^...
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Is the Champernowne sequence polynomially normal?
Let $s:\mathbb{N}\to\{0,1\}$ be the Champernowne sequence, starting with $$0\, 1\, 10\, 11\, 100\, 101\, 110\,\ldots$$
It is well known that this sequence is normal.
Question. If $p:\mathbb{N}\to\...
- 54.6k
24
votes
1
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A mysterious recurrence for primes
For any positive integer $n$, define $s(n)$ as the smallest positive integer $m$ such that the $n$ distinct numbers
$$ (p_1-1)^2,\ (p_2-1)^2,\ \ldots,\ (p_n-1)^2$$
are pairwise incongruent modulo $m$,...
- 18.1k
3
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On the set $\{\sum_{k=1}^n p_k:\ n = 1,2,3,\ldots\}$
For any positive integer $n$, let $S(n)$ be the sum of the first $n$ primes. Then
$$S(1) = 2,\ S(2)=2+3=5,\ S(3)=2+3+5 =10,\ S(4) = 2+ 3+5+7 =17.$$
By the Prime Number Theorem,
$$S(n)\sim \frac{n^2}2\...
- 18.1k
0
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0
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Are there a finite numbers of zeros in this integer sequence?
Consider the triangular array $T(n,k)_{1 \le k \le n}$ defined by the recurrence \begin{align*} T(n,1) &= 1, \\ T(n,k) &= 1+\sum_{i=1}^{k-1} T(n - i, k - 1) -\sum_{i=1}^{n-1} T(n - i, k). \end{...
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1
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Is it true that $\{p_{2^m+1}-p_{2^m}:\ m\in\mathbb Z^+\}=\{2n:\ n\in\mathbb Z^+\}$?
For $n\in\mathbb Z^+=\{1,2,3,\ldots\}$, let $p_n$ denote the $n$th prime. A well known conjecture of de Polignac states that for any $n\in\mathbb Z^+$ there are infinitely many $k\in\mathbb Z^+$ with $...
- 18.1k
4
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1
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295
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On the number of $0$-$1$ vectors with pairwise distinct sums $v_i + v_j$
Let $n \ge 1$. A set of vectors $v_1, \ldots, v_m \in \{0,1\}^n$ is called admissible if all pairwise sums $v_i + v_j$ (with $1 \le i \le j \le m$) are distinct. We want to find the number $a(n)$, ...
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15
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Existence of an odd prime $p$ such that the sum of last two base-$p$ digits is at least $p$
I conjecture that
For every integer $k>78$, there exists an odd prime $p$ such that the sum of last two base-$p$ digits of $k$ is $\geq p$.
We may additionally assume that $k+1$ is a prime, a ...
- 38.6k
2
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1
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388
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Derive homogeneous recurrence from second order one
Let $a,b \in \mathbb{R}$ and sequece $\{f(n)\}_{n=1}^{\infty}$ is given by homogeneous second order recursive relation
$$
f(n):=af(n-1)-b^2f(n-2), \:\:\: n>2
$$
with two arbitrary starting values $...
