1
vote
1answer
86 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
votes
2answers
76 views

Let $u_{n+3} = u_n + 2u_{n+1}$ . Show that $p$ divides $u_p$ for all $p$ prime number.

Let $(u_n)$ a sequence such that $u_0 = 3$, $u_1 = 0$, $u_2 = 4$ and $u_{n+3} = u_n + 2u_{n+1}$ Show that $p$ divides $u_p$ for all $p$ prime number. I'm really stuck on this exercise, ...
1
vote
1answer
58 views

What is the 5000th happy prime number?

Im writing a program that finds the Nth happy prime number. I think it works, but to double check I want to compare what it returns for the 5000th happy prime number. The problem is, I dont know where ...
3
votes
2answers
36 views

What value does the sum of the square of the reciprocals of the prime counting function converge to?

Using my only mathematical tool Wolfram Alpha, I noticed that $\sum_{n=2}^\infty \frac{1}{\pi(x)}$ seems to diverge. Naturally, the entered the following sum and, just like the zeta function, saw that ...
1
vote
1answer
34 views

Finding infinite sequences with pairwise relatively prime outputs.

I am looking for a formula which for every element in $\mathbb{Z}$ as an input, gives pairwise relatively prime outputs. That is for example thanks to Greg Martin's suggestion the positive outputs of ...
4
votes
1answer
124 views

The Sum $\sum_{n=1}^{\infty}\frac{(-1)^{\pi(n)}}{n}$

$$\sum_{n=1}^{\infty}\frac{(-1)^{\pi(n)}}{n}$$ Does this sum converge or does it diverge? Are there any results related to this? ($\pi(n)$ is the number of primes less than or equal to $n$)
37
votes
5answers
3k views

Does every prime divide some Fibonacci number?

I am tring to show that $\forall a \in \Bbb P\; \exists n\in\Bbb N : a|F_n$, where $F$ is the fibonacci sequence defined as $\{F_n\}:F_0 = 0, F_1 = 1, F_n = F_{n-1} + F_{n-2}$ $(n=2,3,...)$. How can ...
11
votes
3answers
269 views

Divergence for $p$ prime numbers and convergence for $m$ composite numbers

Does there exist a sequence $(a_n)_{n\in \mathbb N} \in \mathbb C^{\mathbb N}$ such that : For all $p$ prime numbers the series $\displaystyle \sum_{n\in \mathbb{N}} a_n^p$ diverges, and for ...
4
votes
1answer
61 views

Property of primes / property of other sequences?

Conjecture If we have two consecutive prime numbers $p_{n}$ and $p_{n+1}$, and another arbitrary prime number $p_a$ such that $p_{n} < p_{n+1} < p^2_{a}$, then it follows that $p_{n+1} - ...
1
vote
0answers
62 views

How to get the period of oeis.org/A130166 other than by trail?

oeis.org/A130166 a(0)=1; a(n)=prime(mod(a(n-1),1000)) starts with: ...
2
votes
1answer
115 views

How to prove the convergence of a series of prime numbers

I have a bit of a problem proving that the series: $$ \sum_{p\leq x} \frac{p\ln\left(p\right)}{x^2} $$ where the sum is extended over all prime numbers, converges to 0.5. Any ideas? Thanks in ...
1
vote
1answer
129 views

Primes created by “n + digital-root(n)” sequences

I've looked at the sequences created by repeatedly adding the digital root of a number to the number until it becomes prime. This is the pseudo-code for the program I've used:   n = 0 ...
6
votes
1answer
157 views

The series $2+3x+5x^2+7x^3+11x^4+…$

It occurred to me to ask whether the power series whose coefficients are the primes has non-zero radius of convergence, and if so, what kind of function it produces. Wikipedia has some bounds on the ...
17
votes
2answers
306 views

How to either prove or disprove if it is possible to arrange a series of numbers such the sum of any two adjacent number adds up to a prime number

I'm wondering if it's possible to write a theorem to prove or disprove the possibility of arranging a sequence of numbers (1,2,...n) such that the sum of any two numbers adds up to a prime number. An ...
0
votes
1answer
36 views

Summation of a curious series-repeated division by primes

I am interested in knowing if there is some closed form/formula for the following series: ...
14
votes
2answers
176 views

Conjecture: the sequence of sums of all consecutive primes contains an infinite number of primes

Starting from 2, the sequence of sums of all consecutive primes is: $$\begin{array}{lcl}2 &=& 2\\ 2+3 &=& 5 \\ 2+3+5 &=& 10 \\ 2+3+5+7 &=& 17 \\ ...
1
vote
3answers
135 views

Are there more integers then prime numbers?

I kind of feel this question may have been asked in some way before, but I could not find it. I know there are infinite prime numbers (because Euclid tells us), and there are infinite integers. For ...
2
votes
2answers
81 views

Does there exist an infinite sequence $p_0,p_1,p_2…$ of prime numbers such that $p_k=4 p_{k-1}\pm 1$

$k \in Z^+$ firstly we know that there exists infinetly many primes of the form $4n+1$ by FTA also we see that if we consider finite primes say to $n$ then the recursive formular can be expressed ...
9
votes
2answers
120 views

What are Green's almost primes?

In a general-audience talk, Ben Green explains his famous proof with Terence Tao as an application of Szemerédi's theorem, but placing the primes within a smaller set of almost-primes in which they ...
1
vote
1answer
61 views

The progression $4n+3$ and primes.

Consider an arithmetic sequence $4n+3$. This sequence contains infinitely many primes and infinitely many composites. It is clear that there cannot be $3$ consecutive primes in the sequence as every ...
0
votes
0answers
78 views

Infinite sum over primes

I want to compute the following sum over primes: $$\sum\limits_{p \text{ prime}}\sum\limits_{k=1}^\infty(\log(p^k))\left(\frac{1}{2p^k} - \Phi[-1,1,p^k]\right),$$ where $\Phi[z,s,a]$ is the ...
14
votes
1answer
166 views

Sum of reciprocals of primes factorial: $\sum_{p\;\text{prime}}\frac{1}{p!}$

The series $$\sum_{p\;\text{prime}}\frac{1}{p}=\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\frac{1}{11}+\cdots$$ diverges as is well known. How about the following? ...
0
votes
1answer
44 views

Motivation for Catalan's Aliquot Sequence Conjecture

Catalan's Aliquot Sequence Conjecture states that every aliquot sequence ends in a prime, a repeating aliquot sequence (a set of sociable numbers) or a perfect number. In my naive mind I can't help ...
4
votes
0answers
54 views

Is this a recurrence for the characteristic sequence of composite numbers?

The characteristic sequence of composite numbers is equal to 1 if $n$ is not a prime number and equal to 0 if $n$ is a prime number, starting: $$1, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1,...$$ where the ...
2
votes
1answer
33 views

Need help to prove that $n \geq 3 \implies q_n = p_{n+1}$

Consider $q_n$ such that $$q_n = \sum_{k=2}^{2^n} k \left \lfloor \frac{1}{1 + \text{Abs} \left (n-\left \lfloor \frac{1}{a(k)} \right \rfloor b (k) \right )} \right \rfloor$$ where $$a(x) = ...
0
votes
2answers
101 views

Nonexistence of Limit of Sum of Prime Factors

In trying to prove the following problem, I find great difficulty in proceeding to generalizing some results: Let $s(n)$ be the sum of prime factors of an integer $n$. Prove that $\lim_{n \to \infty} ...
0
votes
0answers
125 views

Value of sum with primes

Can anyone tell me the value of sum $\sum_p\left(\log p(\frac{1}{2p}-\psi(\frac{p+2}{2})+\psi(\frac{p+1}{2})\right)-\sum_n\frac{(\log(2^n)}{2^n}$ where $p$ ranges over prime powers and $n$ ranges from ...
3
votes
1answer
159 views

Calculation involving $\int_2^x \frac{dx}{\log x}$

Background (skip to the gray if you prefer). In Legendre's 1798 work on number theory he conjectured that $\pi(x)\sim \frac{x}{\log x - A}$ in which he proposed that $A = 1.08366.$ Gauss disputed the ...
6
votes
0answers
58 views

Infinite series with prime number [duplicate]

I know the $\sum_{n=1}^\infty \frac{1}{\text{Prime[$n$]}}$ does not converge, but what about the following series? $$\sum_{n=1}^\infty\frac{1}{\text{Prime[Prime[$n$]]}}$$ (Where $\text{Prime[$n$]}$ ...
1
vote
1answer
78 views

Sum containing primes

Can anybody compute the value of $$\sum_p\sum_{k=2}^\infty\frac{\log(p^k)}{k}-\sum_p\sum_{k=2}^\infty\frac{\sum\limits_{p^n<k}\log(p^n)}{k(k+1)}$$ I have tried a lot but cannot think about the ...
1
vote
2answers
184 views

“Interesting” Sequences

Well, here's a question i myself made up and i thought it's interesting if i share it with everyone. We call a sequence of natural numbers (for example $a$) Interesting if (all three must be true): ...
0
votes
1answer
80 views

Discrepancy between terms of sum and sum

My question is why the following happens, and whether we can correct (2) below to account for an errant factor of 2. By a slight generalization* of the argument of this problem we have I think that ...
1
vote
1answer
88 views

Series equivalent to $\sum p_k$

Looking at a theorem of Chebyshev, I noticed that $$\sum_{n=0}^{\infty} \sum_{p_k < n} \frac{(\log p_k)^n}{n!} = 2 + 3 + ...+ p_k.$$ Proof. Letting $x = \log p_k$ and writing out the expansion of ...
1
vote
2answers
104 views

Does $f(n)\sim g(n)$ imply $\lim_{k\to\infty} \frac{1}{k} \sum_n f(n)/g(n) = 1$?

Is it true that $$\lim_{k\to\infty}\frac{1}{k}\sum_{n=1}^k \frac{f(n)}{g(n)} = 1 \leftrightarrow f(n)\sim g(n).$$ My thought: $f(n)\sim g(n) \rightarrow \frac{1}{k}\sum \frac{f(n)}{g(n)} = 1$ since ...
2
votes
1answer
151 views

Limit of $\sum\frac{1}{p(\pi(n))}$

Let $p(n)$ be the nth prime and $\pi(n)$ the number of primes not exceeding n. I wonder if we can show that $$\tag{1} S = \sum_{n= 2}^k \frac{1}{ p (\pi (n))} \sim \log k. $$ We know by comparison ...
6
votes
3answers
208 views

The ordinary generating function for $ζ(s)$

$$\zeta(s)^m = \sum_{n=1}^{\infty} \frac{a_n}{n^s}$$ where $ζ(s)$ is the Riemann zeta function has the ordinary generating function: $$\sum \limits_{n=1}^{\infty} a_nx^n = x + {m \choose 1}\sum ...
2
votes
0answers
171 views

Prime number finding via polynomials

I try to find approximation polynomial to estimate which number is prime or not. Addtion to this, (If It is possible) To find the closed form of coefficients of the series ($c(n)$) Euler found the ...
2
votes
1answer
125 views

Confused about harmonic series and Euler product

So Euler argued that $$1 + \frac{1}{2} + \frac{1}{3} + \frac {1}{4} + \cdots = \frac {2 \cdot 3 \cdot 5 \cdot 7 \cdots} {1 \cdot 2 \cdot 4 \cdot 6 \cdots} $$ which you can rearrange to $$ \left( \frac ...
15
votes
1answer
440 views

How can we prove $\pi =1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\cdots\,$?

I saw the beautiful result that was proved by Euler in Wikipedia but I do not know how it can be proved. $$\pi =1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} - \frac{1}{5} + \frac{1}{6} + ...
13
votes
2answers
372 views

What is the value of $\sum\limits_{i=1}^\infty\frac{1}{p_{p_i}}$ where $p_{i}$ is the $i$th prime?

What is the value of $\sum\limits_{i=1}^\infty\dfrac{1}{p_{p_i}}$ where $p_n$ is the nth prime (and so $p_{p_n}$ is the $k$th prime, where $k$ is the $n$th prime) ? Thus ...
0
votes
1answer
65 views

2 questions on logarithms of $1\over x$ where $x = 10^k -1$

Let $a_n$ denote the logarithm of the $n$'th sum of $\{{1\over9},{1\over99},{1\over999},...\}$, such that $a_n = ...
1
vote
3answers
43 views

Proving $\prod _{k=j}^n \frac{p_{k+1}}{p_k} = \frac{p_{n+1}}{p_j}\!\!,\;\;1\le j\!<\!n$

Let $p_n$ denote the $n$th prime number. How could one prove that: $$\prod \limits_ {k=j}^n \frac{p_{k+1}}{p_k} = \frac{p_{n+1}}{p_j}\!\!,\;\;1\le j\!<\!n$$ Examples: ...
13
votes
3answers
370 views

$\sum\limits_{\text{prime }p} 2^{-p}$ is an irrational number

I need help to prove the following result. $\displaystyle\sum_{\text{prime }p} 2^{-p}$ is an irrational number.
0
votes
4answers
104 views

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1} \frac{1}{p_n}$ convergent? [duplicate]

Let ${P_n}$ be the sequence of all consecutive prime numbers. Is $\sum_{n\geq 1}\frac{1}{p_n}$ convergent?
21
votes
1answer
263 views

Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$?

Is any closed-form representation known for the sum $\sum\limits_{n=1}^{\infty}\frac{\mu(n)\log n}{n^2}$, where $\mu(n)$ is the Möbius $\mu$-function?
0
votes
1answer
64 views

General term of this sequence

I wanted to know the General term or the function to generate this sequence I found on OEIS. It is the number of ways to express $2n+1$ as $p+2q$; where $p$ and $q$ can be odd prime number and even ...
4
votes
1answer
206 views

What is the smallest positive common difference of a 6-term arithmetic progression consisting entirely of (positive) prime numbers?

What is the smallest positive common difference of a 6-term arithmetic progression consisting entirely of (positive) prime numbers? are divisibility rules applicable here?
4
votes
3answers
832 views

Is there a rule for prime numbers?

I've passed by this article: http://gauravtiwari.org/2011/12/11/claim-for-a-prime-number-formula/ and this paper: http://www.m-hikari.com/ams/ams-2012/ams-73-76-2012/kaddouraAMS73-76-2012.pdf They ...
7
votes
1answer
72 views

Proving $\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$?

$$\lim_{n\to\infty}\left(n-\sum_{k=2}^{n}\frac{1}{\sum_{i=1}^{\infty}\frac{1}{i^k}}\right)=1+\sum_{p\in P}\frac{1}{p\left(p-1\right)}$$ $P$ is primes. Interesting question ran across while tutoring. ...
6
votes
1answer
94 views

Testing for convergence $\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$

How would we test for convergence the series below? $$\sum_{j=1}^{\infty}\frac{1}{\sum_{i=1}^{j}p_i}$$ where $p_i$ is the $i$th prime number. I'd be glad to learn an elementary way. Thanks.