All Questions
Tagged with prime-factorization sequences-and-series
37
questions
2
votes
1
answer
86
views
Show that for an odd integer $n ≥ 5$, $5^{n-1}\binom{n}{0}-5^{n-2}\binom{n}{1}+…+\binom{n}{n-1}$ is not a prime number.
I would prefer no total solutions and just a hint as to whether or not I’m at a dead end with my solution method.
So far this is my work:
From the binomial expansion,
$$\sum_{j=0}^n 5^{n-j}(-1)^j\...
- 183
5
votes
2
answers
194
views
Periodic sequences of integers generated by $a_{n+1}=\operatorname{rad}(a_{n})+\operatorname{rad}(a_{n-1})$
Let's define the radical of the positive integer $n$ as
$$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\ p\text{ prime}}}p$$
and consider the following Fibonacci-like sequence
$$a_{n+1}=\...
- 1,828
3
votes
1
answer
402
views
Consider $A$ the set of natural numbers with exactly 2019 divisors
Consider $A$ the set of natural numbers with exactly 2019 divisors, and for each $n \in A$ denote
$$
S_n = \frac{1}{d_1+\sqrt{n}} + \frac{1}{d_2+\sqrt{n}} + ... + \frac{1}{d_{2019}+\sqrt{n}},
$$
...
- 61
3
votes
1
answer
114
views
What comes next in this sequence $1,4,8,16,17,44,58,76,\dots$? [closed]
My friend John asked me what is the next number in the sequence
$$1,4,8,16,17,44,58,76,\dots ?$$
I told him it could be anything, and I asked for a clue. He told me it involves adding numbers that ...
user913631
-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
3
votes
0
answers
88
views
Do the sequences defined by $a_n=a_{n-1}+(\text{the least prime factor of }a_{n-1})+1$ starting with $2,6,14,\ldots$ merge?
Let $S_k$ be the sequence defined by $a_k(1)=k,\ a_k(n)=a_k(n-1)+(\text{the least prime factor of }a_k(n-1))+1$.
A diagram of these sequences for around $k<100$ is shown below. As you can see, $S_3$...
- 1,451
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:
$$a_n\text{ is the smallest product of }n\text{ distinct odd primes, }m=p_1p_2\dots p_n\text{, such that }\frac{d+\frac{m}{d}}{...
- 1,282
2
votes
1
answer
110
views
If $p$ and $q$ are coprime positive integers s.t. $\frac{p}{q}=\sum_{k=0}^{100}\frac1{3^{2^k}+1}$, what is the smallest prime factor of $p$?
If the sum $$S=\frac14+\frac1{10}+\frac1{82}+\frac1{6562}+\cdots+\frac1{3^{2^{100}}+1}$$
is expressed in the form $\frac pq,$ where $p,q\in\mathbb N$ and $\gcd(p, q) =1.$ Then what is smallest prime ...
0
votes
1
answer
67
views
Is the sequence infinite? Finite? Is there a general formula to determine n th term?
Sequence of numbers whose factorial on prime factorisation contains prime powers of prime numbers, whose power is greater than $1$ or contains multiplicity of one for all prime numbers less than equal ...
1
vote
2
answers
86
views
To determine multiplicity of $2$ in $n!$ [duplicate]
Is there a general formula for determining multiplicity of $2$ in $n!\;?$
I was working on a Sequence containing subsequences of 0,1. 0 is meant for even quotient, 1 for odd quotient.
Start with k=3,...
4
votes
2
answers
199
views
Lower bound related to the number of distinct prime numbers
Let $\omega(n)$ be the number of distinct prime factors of $n$ (without multiplicity, of course). I know some results about average of $\omega(n)$. But I didn't find any result about the following: ...
- 629
1
vote
1
answer
488
views
How to solve Shonk Sequences?
A Shonk sequence is a sequence of positive integers in which
each term after the first is greater than the previous term, and
the product of all the terms is a perfect square
For example: 2, 6, 27 ...
13
votes
1
answer
344
views
Prime divisors of the sequence terms $a_n=a\cdot 2017^n+b\cdot 2016^n$
I am dealing with the test of the OBM (Brasilian Math Olimpyad), University level, 2017, phase 2.
As I've said at another topic (question 1), I hope someone can help me to discuss this test.
The ...
- 1,525
-2
votes
1
answer
107
views
On an inequality involving the radical of an integer and its greatest prime factor
Let $n\geq 1$ an integer, in this post I denote the greatest prime dividing $n$ as $\operatorname{gpf}(n)$, and the product of the distinct prime numbers dividing $n$ as $$\operatorname{rad}(n)=\prod_{...
user243301
1
vote
1
answer
243
views
Square-free integers in the sequence $n^{\operatorname{rad}(n)}+\operatorname{rad}(n)+1$
For integers $n\geq 1$ in this post we denote the square-free kernel as $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p\text{ prime}}}p,$$ that is the product of distinct primes dividing an ...
user243301
0
votes
1
answer
163
views
On prime-perfect numbers and the equation $\frac{\varphi(n)}{n}=\frac{\varphi(\operatorname{rad}(n))}{\operatorname{rad}(\sigma(n))}$
While I was exploring equations involving multiple compositions of number theoretic functions that satisfy the sequence of even perfect numbers, I wondered next question (below in the Appendix I add a ...
user243301
0
votes
1
answer
166
views
Natural density of integers such that $\forall 1\leq k\leq n-1$ satisfy $\operatorname{rad}(k)+\operatorname{rad}(n-k)\geq \operatorname{rad}(n)$
For an integer $n>1$ we defined its square-free kernel $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p \text{ prime}}}p$$
as the product of distinct prime factors dividing it, with the ...
user243301
3
votes
1
answer
106
views
A problem similar than that of the amicable pairs using the function $\operatorname{rad}(k)$: a first statement or conjecture
In this post we denote the product of distinct primes dividing an integer $k> 1$ as $\operatorname{rad}(k)$, with the definition $\operatorname{rad}(1)=1$, that is the so-called radical of an ...
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
2
votes
1
answer
68
views
On miscellaneous questions about perfect numbers II
Let $\varphi(m)$ the Euler's totient function and $\sigma(m)$ the sum of divisors function. We also denote the product of primes dividing an integer $m>1$ as $\operatorname{rad}(m)$, that is the ...
user243301
2
votes
1
answer
162
views
On even integers $n\geq 2$ satisfying $\varphi(n+1)\leq\frac{\varphi(n)+\varphi(n+2)}{2}$, where $\varphi(m)$ is the Euler's totient
This afternoon I am trying to get variations of sequences inspired from the inequality that defines the so-called strong primes, see the definition of this inequality in number theory from this ...
user243301
2
votes
1
answer
105
views
Square-free integers in the sequence $\lambda+\prod_{k=1}^n(\varphi(k)+1)$, where $\lambda\neq 0$ is integer
While I was exploring the squares in the sequence defined for integers $n\geq 1$
$$\prod_{k=1}^n(\varphi(k)+1),\tag{1}$$
where $\varphi(m)$ denotes the Euler's totient function I wondered a different ...
user243301
0
votes
2
answers
72
views
On variations of Rowland's sequence using the radical of an integer $\prod_{p\mid n}p$
This afternoon I tried to create a Rowland's sequence using the radical of an integer in my formula. I don't know if it was in the literature, but I know that also there were variations on Rowland's ...
user243301
2
votes
0
answers
48
views
Compare $\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\rfloor\right)}$ and $\sum_{k\mid n}k^{\operatorname{rad}\left(\frac{n}{k}\right)}$
I would like to know how do a comparison between the sizes of these functions defined for integers $n\geq 1$, when $n$ is large
$$f(n):=\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\...
user243301
2
votes
0
answers
64
views
Find next larger number with same number of prime factors
Is there are way to determine, given a (composite) number $n$ and a list of its prime factors, the next larger (composite) number with the same number of prime factors? Clarification: Not the same ...
- 305
4
votes
1
answer
168
views
Does a sequence based on hereditary factorisation always terminate?
The well-known Goodstein sequences are based on the hereditary base-$b$ notation, where you don't just present the digits in base $b$, but also the corresponding exponents etc.
That lead me to the ...
- 1,610
31
votes
0
answers
1k
views
Have I discovered an analytic function allowing quick factorization?
So I have this apparently smooth, parametrized function:
The function has a single parameter $ m $ and approaches infinity at every $x$ that divides $m$.
It is then defined for real $x$ apart from ...
- 1,091
4
votes
1
answer
194
views
Sequence generated by $2^k-1$ contains new prime factors
I was playing around with the sequence where the $k^{th}$ number is equal to $2^k-1$. It seems that all numbers except $63$ contain at least one new prime in there prime factorization. That is a prime ...
- 2,086
3
votes
0
answers
950
views
A very nice pattern involving prime factorization
A while ago I was fiddling around with prime numbers and C++. I defined:
$$f_a(b)= \text{ the amount of numbers } 2^a\leq n<2^{a+1}\text{ with } b \text{ prime factors}$$
I calculated $f_a(b)$ for ...
- 8,421
2
votes
1
answer
203
views
Do we know the rate of divergence of the sum of reciprocals of the $k$-almost primes?
A $k$-almost prime is a positive integer having exactly $k$ prime factors, not necessarily distinct. Let $\mathbb{P}_k$ be the set of the $k$-almost primes and let $$ \rho_k(n):=\sum\limits_{\substack{...
- 7,299
0
votes
2
answers
44
views
Series of positive factors of a number divided by that number
Let $S_n$ be the sum of the positive factors of $2015^n$, with $n$ being a positive integer approaching infinity. What is $\dfrac{S_n}{2015^n}$?
I might be on the wrong track, but I figure that if $x ...
- 1
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
9
votes
2
answers
840
views
Is 641 the Smallest Factor of any Composite Fermat Number?
Consider the sequence $a_n = 2^{2^n}+1$ of so-called Fermat numbers. It's well known that $a_5$ isn't prime ($a_5 = 641 \cdot 6700417$, this is due to Euler). What I want to know about this sequence ...
- 2,195
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
0
votes
1
answer
1k
views
Sum of number of factors of first N numbers [duplicate]
Given a number N ( Value can be large like N < 10^9 ) How can we calculate sum of the number of factors of first N numbers??
Example :
For n = 3
Answer:
= #f(1) + #f(2) + #f(3) --- { #f(n) ->...
- 21