All Questions
12
questions
5
votes
2
answers
194
views
Periodic sequences of integers generated by $a_{n+1}=\operatorname{rad}(a_{n})+\operatorname{rad}(a_{n-1})$
Let's define the radical of the positive integer $n$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$
and consider the following Fibonacci-like sequence
$$a_{n+1}=\...
- 1,828
-1
votes
2
answers
59
views
Prove that series $\sum_{n\in M(P)} \frac{1}{n}$ converges and find its sum [closed]
Let $P = \{p_1, p_2, \ldots, p_k\}$ be a finite set of prime numbers, and $M(P)$ be a set of natural numbers, whose prime divisors are in $P$. How can I prove that $$\sum_{n\in M(P)} \frac{1}{n}$$ ...
- 1
1
vote
1
answer
279
views
What's Special about Rowland’s Prime-Generating Sequence?
Recently I asked this question and quickly got back some excellent responses.
I asked the question because I came across a paper by Eric Rowland called "A Natural Prime-Generating Recurrence"...
- 692
2
votes
0
answers
51
views
What is the smallest product, $m$, of $6$ distinct odd primes such that $\frac{d+\frac{m}{d}}{2}$ is prime for all $d$ dividing $m$?
I am currently working on a sequence, $a_n$, that is defined as follows:
$$a_n\text{ is the smallest product of }n\text{ distinct odd primes, }m=p_1p_2\dots p_n\text{, such that }\frac{d+\frac{m}{d}}{...
- 1,282
-2
votes
1
answer
107
views
On an inequality involving the radical of an integer and its greatest prime factor
Let $n\geq 1$ an integer, in this post I denote the greatest prime dividing $n$ as $\operatorname{gpf}(n)$, and the product of the distinct prime numbers dividing $n$ as $$\operatorname{rad}(n)=\prod_{...
user243301
5
votes
1
answer
135
views
What about sequences $\{\sum_{k=1}^n (\operatorname{rad}(k))^p\}_{n\geq 1}$ containing an infinitude of prime numbers, where $p\geq 1$ is integer?
We denote the radical of the integer $n> 1$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p,$$
taking $\operatorname{rad}(1)=1$ that is this definition from Wikipedia.
In ...
user243301
4
votes
1
answer
168
views
Does a sequence based on hereditary factorisation always terminate?
The well-known Goodstein sequences are based on the hereditary base-$b$ notation, where you don't just present the digits in base $b$, but also the corresponding exponents etc.
That lead me to the ...
- 1,610
31
votes
0
answers
1k
views
Have I discovered an analytic function allowing quick factorization?
So I have this apparently smooth, parametrized function:
The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$.
It is then defined for real $x$ apart from ...
- 1,091
3
votes
0
answers
950
views
A very nice pattern involving prime factorization
A while ago I was fiddling around with prime numbers and C++. I defined:
$$f_a(b)= \text{ the amount of numbers } 2^a\leq n<2^{a+1}\text{ with } b \text{ prime factors}$$
I calculated $f_a(b)$ for ...
- 8,421
3
votes
0
answers
107
views
Special $\omega(n)$-sequence
Let $k$ be a natural number, $\omega(n)$ the number of distinct prime factors of $n$.
The object is to find a number $n$ with $\omega(n+j)=j+1$ for each $j$ with
$0\le j\le k-1$. In other words, a ...
- 85.1k
3
votes
0
answers
105
views
Do there exist any cycles for these number sequences?
We define, for $k\in\mathbb{N}$, the sequence $\left(S_{k,n}\right)_{n\in\mathbb{N}}$:
$$S_{k,1}=k,\;\;\; S_{k,n+1}=p_1q_1\cdots p_mq_m \text{ (written out in decimal)}$$
Where $p_1^{q_1}*\cdots *p_m^{...
- 3,540
1
vote
1
answer
128
views
Is $(1+2+3+…)=(1+2+2^2+2^3+…)(1+3+3^2+…)(1+5+5^2+…)…$?
Are these equal?
$$(1+2+3+…)=(1+2+2^2+…)(1+3+3^2+…)(1+5+5^2+…)…$$
Where the RHS has a series for each prime. Looks like they are the same series by the fundamental theorem of arithmetic.
Every number ...
- 3,115