All Questions
Tagged with collatz-conjecture sequences-and-series
9 questions
5
votes
0
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363
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A Collatz-like map?
Consider the map $\psi$ acting on triples $(a\leq b\leq c)$ of three positive natural integers with $\mathrm{gcd}(a,b,c)=1$ as follows:
Set $$(a',b',c')=\left(\frac{a}{\mathrm{gcd}(a,bc)},\frac{b}{\...
- 17.9k
16
votes
3
answers
4k
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Is it known that the Collatz-like sequence with 7n+1 diverges to infinity starting with 7?
In this question I was wondering if the $3$ in the Collatz conjecture is arbitrary, and when I wrote that question I tried to change to $7n+1$ starting with the seed number $7$, the sequence appears ...
- 541
0
votes
1
answer
259
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Finding a strictly increasing Collatz sequence of arbitrary length [closed]
Is there a formula to construct a Collatz (3x + 1) sequence of arbitrary length that is strictly increasing? Obviously one can do this with a strictly decreasing sequence by just taking $2^n$ but I ...
1
vote
0
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435
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Has the Collatz been investigated as a recursive function?
Does anyone ever write the Collatz conjecture as a single algebraic, recursive sequence? For example, a crude version might be:
$$
g(n+1)=\delta _{1,g(n)}+(1-\delta _{1,g(n)})*\left(\left(\frac{cos(\...
- 29
1
vote
1
answer
524
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What is the quotient (pseudo)metric $d_\sim$ and how do I identify the infimum of possible sequences in this instance?
Let $Z$ be the the set of dyadic and ternary rationals in the interval $\left[\frac12,1\right)$ whose 3-adic valuation is either $-1$ or $0$, with the standard absolute value topology inherited from ...
- 308
5
votes
2
answers
1k
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Proof that $3^ns + \sum_{k=0}^{n-1} 3^{n-k-1}2^{a_k}=2^m.$
How would I go about proving the following:
For any odd positive integer $s$, there exists a sequence of nonnegative integers $( a_0, a_1, \cdots, a_{n-1})$ and a nonnegative integer $m$ such that,
$...
-1
votes
2
answers
370
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Are there integral solutions for $(2a-1)(2^{(b+c)}-3^c )=2^b-1$?
Can anyone prove this assertion? Or at least suggest a method of attack? It has come up in my research.
There do not exist $a,b$ and $c$ such that$$
(2a-1)(2^{(b+c)}-3^c )=2^b-1
$$where $a>0,b&...
17
votes
2
answers
1k
views
Does 53 diverge to infinity in this Collatz-like sequence?
This function has been explored a bit at MSE (in June 2016):
\begin{eqnarray}
f(n) &=& (n-1)^2 \; \textrm{if} \; (n \bmod 4) = 1\\
f(n) &=& \lfloor n/4 \rfloor \; \textrm{otherwise}
\...
- 151k
7
votes
1
answer
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Beyond Collatz: A $5n+1$ conjecture? [closed]
Let
$$x_{n+1} = \begin{cases} x_n/2 &;\text{if } x_n \equiv 0 \pmod{2}\\ k\,x_n+1 &; \text{if } x_n\equiv 1 \pmod{2} \end{cases}$$
and $k=3$ and $x_n\in\Bbb N$. Collatz conjectured for this ...
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