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,
$$\sum_{j=0}^n 5^{n-j}(-1)^j\...
- 183
5
votes
2
answers
192
views
Periodic sequences of integers generated by $a_{n+1}=\operatorname{rad}(a_{n})+\operatorname{rad}(a_{n-1})$
Let's define the radical of the positive integer $n$ as
$$\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
401
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
273
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}$$
is expressed in the form $\frac pq,$ where $p,q\in\mathbb N$ and $\gcd(p, q) =1.$ Then what is smallest prime ...
0
votes
1
answer
67
views
Is the sequence infinite? Finite? Is there a general formula to determine n th term?
Sequence of numbers whose factorial on prime factorisation contains prime powers of prime numbers, whose power is greater than $1$ or contains multiplicity of one for all prime numbers less than equal ...
1
vote
2
answers
86
views
To determine multiplicity of $2$ in $n!$ [duplicate]
Is there a general formula for determining multiplicity of $2$ in $n!\;?$
I was working on a Sequence containing subsequences of 0,1. 0 is meant for even quotient, 1 for odd quotient.
Start with k=3,...
4
votes
2
answers
198
views
Lower bound related to the number of distinct prime numbers
Let $\omega(n)$ be the number of distinct prime factors of $n$ (without multiplicity, of course). I know some results about average of $\omega(n)$. But I didn't find any result about the following: ...
- 629
1
vote
1
answer
488
views
How to solve Shonk Sequences?
A Shonk sequence is a sequence of positive integers in which
each term after the first is greater than the previous term, and
the product of all the terms is a perfect square
For example: 2, 6, 27 ...
13
votes
1
answer
342
views
Prime divisors of the sequence terms $a_n=a\cdot 2017^n+b\cdot 2016^n$
I am dealing with the test of the OBM (Brasilian Math Olimpyad), University level, 2017, phase 2.
As I've said at another topic (question 1), I hope someone can help me to discuss this test.
The ...
- 1,525
-2
votes
1
answer
107
views
On an inequality involving the radical of an integer and its greatest prime factor
Let $n\geq 1$ an integer, in this post I denote the greatest prime dividing $n$ as $\operatorname{gpf}(n)$, and the product of the distinct prime numbers dividing $n$ as $$\operatorname{rad}(n)=\prod_{...
user243301