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Results tagged with nt.number-theory
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user 231922
Prime numbers, diophantine equations, diophantine approximations, analytic or algebraic number theory, arithmetic geometry, Galois theory, transcendental number theory, continued fractions
2
votes
0
answers
57
views
Vector of only ones by activating the bits
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 …
- 3,018
1
vote
1
answer
109
views
Numbers $m$ for which coefficients of the polynomial $p(m,x)$ are relatively prime
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 …
- 3,018
0
votes
1
answer
87
views
Property of composite numbers
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) …
- 3,018
0
votes
1
answer
161
views
Number of positive integers $k$ such that there exists a nonnegative integer $m$ with $k + k...
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
$$0, 1, 1, 2, 1, 3, 1, 2, 1, 3, 1, 3, 1, 2, 1, 2, 1, 3, 1, 3$ …
- 3,018
2
votes
1
answer
238
views
Sequence that sums up to INVERTi transform applied to the ordered Bell numbers
$\DeclareMathOperator\wt{wt}$Let $\wt(n)$ be A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$).
Let $f(n)$ be A007814, the exponent of the highest power of $2$ dividin …
- 3,018
6
votes
2
answers
350
views
Sequence of $k^2$ and $2k^2$ ordered in ascending order
Let $\eta(n)$ be A006337, an "eta-sequence" defined as follows:
$$\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 …
- 3,018
7
votes
1
answer
670
views
One conjecture by sequencedb.net
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 …
- 3,018
3
votes
0
answers
96
views
Identical digits at the end of adjacent terms of the sequence
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 $n$. …
- 3,018
3
votes
1
answer
280
views
Sum with products turned into subsequences
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
$$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots) …
- 3,018
3
votes
1
answer
232
views
Sum with Stirling numbers of the second kind
Let $wt(n)$ be A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$)
and
$$n=2^{t_1}(1+2^{t_2+1}(1+\dots(1+2^{t_{wt(n)}+1}))\dots)$$
Then we have an integer sequence given …
- 3,018
2
votes
1
answer
286
views
Difference between $n$-th and $(n-1)$-th composite numbers
Let $f(n)$ = 1 if $n$ belongs to A014689, $\operatorname{prime}(n)-n$, the number of nonprimes less than $\operatorname{prime}(n)$. Here $\operatorname{prime}(n)$ is the $n$-th prime number, $\operato …
- 3,018
0
votes
1
answer
89
views
$a(16n+k)=b(16n+k)-c(16n)$ for $n\geqslant0$, $0 < k < 16$ where $c(n)=b(n)-a(n)$
Let $a(n)$ be A339970 = A329697$($A019565$(2n))$: the sequence begins with
$$0, 1, 1, 2, 2, 3, 3, 4, 2, 3, 3, 4, 4, 5, 5, 6, 2, 3, 3, 4, 4$$
Also let's consider
$$\ell(n)=\left\lfloor\log_{2}(n)\right …
- 3,018
1
vote
0
answers
55
views
Subsequences related with square table
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 …
- 3,018
1
vote
0
answers
40
views
Permutation of nonnegative integers applied to the numbers $n$ whose binary expansion does n...
Let $\operatorname{wt}(n)$ is A000120, number of $1$'s in binary expansion of $n$ (or the binary weight of $n$) and
$$n=2^{b_1}(1+2^{b_2+1}(1+2^{b_3+1}(1+\cdots(1+2^{b_{\operatorname{wt}(n)-1}+1}(1+2^ …
- 3,018
2
votes
2
answers
156
views
Modulo $2$ binomial transform of $m^n$
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 …
- 3,018