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Take any binary vector of length $4n+1$ with $n\geqslant0$. We can activate any bits. When a bit is activated, neighboring bits change their values $0$ -> $1$, $1$ -> $0$. Our goal is to turn the orig …

Let $\eta(n)$ be A006337, an "eta-sequence" defined as follows: $$\eta(n)=\left\lfloor(n+1)\sqrt{2}\right\rfloor-\left\lfloor n\sqrt{2}\right\rfloor$$ Sequence begins $$1, 2, 1, 2, 1, 1, 2, 1, 2, 1, 1 …

$\DeclareMathOperator\wt{wt}$Let $\wt(n)$ be A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividin …

From A248667: The polynomial $p(n,x)$ is defined as the numerator when the sum $$1 + \frac{1}{nx + 1} + \frac{1}{(nx + 1)(nx + 2)} + \cdots + \frac{1}{(nx + 1)(nx + 2)\cdots(nx + n - 1)}$$ is written …

Let $a(n)$ be the number of positive integers $k$ such that there exists a nonnegative integer $m$ with $k + k^m = n$. The sequence begins $$0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3$ …

Let $a(n)$ be A000041 i.e. the number of partitions of $n$ (the partition numbers). Let $T(n, k)$ be A083906. Here $$ T(n, k) = [q^k]\sum\limits_{m=0}^{n} \binom{n}{m}_q $$ where $\binom{n}{m}_q$ …

Let $a(n)$ be A214973, number of terms in greedy representation of $n$ using Fibonacci and Lucas numbers. Let $b(n)$ be A329320, sequence which arises from attempts to simplify computing of A329319. H …

Let $a(n)$ be the sequence of composite numbers (starting from $4$). Let $$b(n)=a(n-1)a(n-2) \operatorname{mod} a(n)$$ Obviously, $b(1)=b(2)=0$. I conjecture that with the only exception for the $b(3) …

Let $wt(n)$ be A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$$ Then we have an integer sequence given …

Let $m\geq 2$ be a fixed integer. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. …

Let $p, q \in \mathbb{Z}$. Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and $$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots) …

Let $\operatorname{wt}(n)$ be A000120, i.e. the number of $1$'s in binary expansion of $n$ (or the binary weight of $n$). Let $a(n)$ be the sequence of numbers $k$ such that $\operatorname{wt}(k)\oper …

Let $m\geqslant1$ be a fixed integer. Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of …

Let $m \in \mathbb{R}$. Let $f(n)$ be A007814, exponent of highest power of $2$ dividing $n$, a.k.a. the binary carry sequence, the ruler sequence, or the $2$-adic valuation of $n$. Let $g(n)$ be A129 …

Let $f(n)$ = 1 if $n$ belongs to A014689, $\operatorname{prime}(n)-n$, the number of nonprimes less than $\operatorname{prime}(n)$. Here $\operatorname{prime}(n)$ is the $n$-th prime number, $\operato …