All Questions
Tagged with integer-sequences permutations
12 questions
10
votes
0
answers
252
views
Permutation of positive integers
Let $a_n$ be a sequence such that $a_1=1$ and for each $n \geq 1$ $a_{n+1}$ is the smallest positive integer distinct from $a_1,a_2,...,a_n$ such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$....
- 3,153
0
votes
0
answers
63
views
Pairs of permutations such that $p(n)<2^k$ iff $n<2^k$
Let $p(n)$ be an arbitrary permutation of natural numbers such that $p(n)<2^k$ iff $n<2^k$.
Let $q(n)$ be an inverse permutation of $p(n)$.
Let
$$
\ell(n)=\left\lfloor\log_2 n\right\rfloor
$$
...
- 4,948
0
votes
1
answer
122
views
Permutation of the natural numbers from operation related to binary expansion of $n$
Let
$$
\ell(n) = \left\lfloor\log_2 n\right\rfloor
$$
Let $T(n,k)$ be a $(k+1)$-th bit from the right side in the binary expansion of $n$. Here
$$
T(n, k) = \left\lfloor\frac{n}{2^k}\right\rfloor \...
- 4,948
2
votes
0
answers
76
views
Uniqueness of the permutation
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
- 4,948
1
vote
0
answers
109
views
Existence of binary permutations with a given property
Let
$$\ell(n)=\left\lfloor\log_2 n\right\rfloor$$
Let
$$f(n)=n-2^{\ell(n)}$$
Let $a(n)$ be a permutation of the nonnegative integers such that $a(0)=0$, $a(n)=n$ if $n$ is a power of $2$ and ...
- 4,948
2
votes
1
answer
172
views
Permutation and its binary analog
Let $f(n)$ be A000045(n), i.e., Fibonacci numbers: $f(n)=f(n-1)+f(n-2)$ for $n>1$ with $f(0)=0$ and $f(1)=1$.
Let $g(n)$ be A072649, i.e., $n$ occurs $f(n)$ times. The sequence begins with
$$1, 2, ...
- 4,948
1
vote
0
answers
81
views
Infiniteness of the pairs of sequences with a given conditions
Let
$$\varphi=\frac{1+\sqrt{5}}{2}$$
Let
$$a_1(n)=\left\lfloor n\varphi \right\rfloor, a_2(n)=n+a_1(n)$$
Let $\operatorname{tr}(n)$ be A007814, i.e., the number of trailing zeros in the binary ...
- 4,948
0
votes
0
answers
61
views
Stolarsky array and Stolarsky representation
Let $T(n,k)$ be A035506, i.e., Stolarsky array read by antidiagonals. Here we consider that $T(n,k)=0$ for $n<1, k<1$.
Let $a(n)$ be A200714, i.e., Stolarsky representation interpreted as binary ...
- 4,948
18
votes
2
answers
992
views
A conjecture harmonic numbers
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:
$a(n)$ ...
- 181
9
votes
0
answers
398
views
When do almost all these invariants of tensors vanish?
Let $A,B,C,D$ be $n$-dimensional vector spaces over a field $k$.
There is a natural homomorphism from the $mn^m$th tensor power $A^{\otimes (m n^m)} $ of $A$ to $k$ given by the determinant map $A^{\...
- 149k
4
votes
1
answer
253
views
Betweenness in permutations
Let us consider permutations $\pi$ on $\{1,\dots,n\}$ as sequences $\pi(1),\pi(2),\dots,\pi(n)$. For a permutation $\pi$ let $R(\pi)$ be the ternary relation with $(x,y,z)\in R(\pi)$ whenever element $...
- 205
14
votes
1
answer
427
views
A Collatz-like question about permutations
An answer to this question would provide an explicit counterexample to this question, but otherwise I don't know if it is interesting.
Consider all permutations $\pi$ on the natural numbers such that ...
- 37.7k