1
vote
0answers
14 views

Alternating series involving sums of k-primes

As an exercise, if $p_k$ are positive integers composed of k primes including repetition and $\pi_k(n)$ the number of $p_k$ not exceeding n can we show that for the alternating series of sums of ...
1
vote
0answers
55 views
+50

About a result concerning Mersenne primes

I want to verify the proof of this result Theorem: If $p>2$ is a prime and $$H_{p}=\frac{(\sqrt3+2)^{2^{p-1}}+1}{(2^{p}-1)(\sqrt 3+2)^{2^{p-2}}}$$ is a natural number then $2^{p}-1$ is a prime ...
0
votes
1answer
30 views

Proving an inequality on the sum of $\log$ of primes.

Let $S(x)=\sum_{p\leq x} \ln(p)$ where $\sum_{p\leq x}$ denotes a summation over the positive prime numbers that are $\leq x$ Prove that $\forall n \in \mathbb N, S(2n+1)-S(n+1)\leq ...
2
votes
1answer
48 views

Divergence of a series containing primes

Is there an easy proof showing that the series $1/p$, where $p$ changes over prime numbers, is divergent?
0
votes
1answer
44 views

How many terms are required to get $D$ digits of Riemann zeta prime function?

How many terms are required to get $D$ digits of Riemann zeta prime function $\zeta_p(s) = \sum_p \frac{1}{p^s}$? Sebah & Gourdon mentions that finding $\zeta_p(2)$ to 20 digits by using $\sum_p ...
4
votes
7answers
110 views

Finite sequences of prime numbers

There is a lot of prime sequences: prime numbers in a special form. For example Mersenne primes are primes of the the form $2^n-1$, or Pythagorean prime are primes of the form $4n+1$. Even primes are ...
3
votes
1answer
87 views

Consecutive Prime Problem

Consecutive primes whose quotient of their product and sum is itself a prime number. $$ 2 \times 3 \times 5 = 30 $$ $$ 30/10 = 3 $$ $$ 3 \times 5 \times 7 = 105 $$ $$ 105/15 = 7 $$ Question: ...
2
votes
1answer
46 views

Is $\sum_{n=1}^N e^{2 \pi p_n z i}$ bounded for irrational $z$?

Let $p_n$ be the $n$th prime number. If $z$ is irrational real, is it known whether the partial sums $\sum_{n=1}^N e^{2 \pi p_n z i}$ are bounded? It seems the partial sums are unbounded if $z$ is ...
10
votes
3answers
374 views

Consecutive Prime Gap Sum (Amateur)

List of the first fifty prime gaps: 1, 2, 2, 4, 2, 4, 2, 4, 6, 2, 6, 4, 2, 4, 6, 6, 2, 6, 4, 2, 6, 4, 6, 8, 4, 2, 4, 2, 4, 14, 4, 6, 2, 10, 2, 6, 6, 4, 6, 6, 2, 10, 2, 4, 2, 12, 12, 4, 2, 4. My ...
2
votes
2answers
183 views

Prime Number Sum Sequence (Amateur)

SOLVED: This is false Beginning with 3, add the next consecutive prime (2) and then take that sum (5) and add that to next consecutive prime (3) to get (8), and so on... $$ 3 + 2 = 5 $$ $$ 5 + 3 = 8 ...
11
votes
0answers
364 views

Sorting of prime gaps

Let $g_i $ be the $i^{th}$ prime gap $p_{i+1}-p_i.$ If we re-arrange the sequence $ (g_{n,i})_{i=1}^n$ so that for any finite $n$ the gaps are arranged from smallest to largest we have a new sequence ...
1
vote
1answer
35 views

Given a sequence, construct a function whose integral is equal to the sum of the sequence

Let $P_n$ be the sequence of prime numbers, where $P_0=2$. Given $m\in\mathbb{N}$, how can we construct $f(x)$ such that: $\displaystyle\forall{0}\leq{i}\leq{m}:f(i)=P_i$ ...
7
votes
2answers
179 views

Does $\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ converge to the golden ratio?

The sum $\displaystyle\sum\limits_{n=2}^{\infty}\frac{1}{n\ln(n)}$ does not converge. But the sum $\displaystyle\sum\limits_{n=1}^{\infty}\frac{1}{P_n\ln(P_n)}$ where $P_n$ denotes the $n$th prime ...
1
vote
0answers
46 views

Sequence of primes by concatenating digits in a given base.

Given a base, $b$ is there is a sequence $\lbrace a_n\rbrace_{n\geq 0}$ where $a_k \in \lbrace 1,2\cdots, b-1\rbrace$so that the sequence: $$b_n:= \sum_{k=0}^n a_kb^k$$ is a sequence of primes ...
4
votes
0answers
64 views

Euler's proof of divergence of sum of reciprocals of primes

On Wikipedia at link currently is: \begin{align} \ln \left( \sum_{n=1}^\infty \frac{1}{n}\right) & {} = \ln\left( \prod_p \frac{1}{1-p^{-1}}\right) = -\sum_p \ln \left( 1-\frac{1}{p}\right) \\ ...
1
vote
1answer
57 views

Limit of Sum $\sum \frac{p(k+1) - p(k)}{p(k)^2}$ ? (corrected)

This is a re-post with my (corrected) earlier typo in red as a reminder. The question is about $$(A)\hspace{10mm}\lim_{n \to \infty} \sum_{k=\color{red}2}^{n} \frac{p(k +1) - p(k)}{p(k)^2} = ...
1
vote
1answer
67 views

Characterizing the primes which don't divide any Pell-Lucas number(s)

For integer $n$, let $P_n$ be a Pell number, and $Q_n$ its companion. Is there a characterization of the prime numbers $p$ which don't divide any $Q_n$? By brute-force search, I found that this ...
2
votes
0answers
89 views

Questions about central polygonal numbers $1, 2, 4, 7, 11, 16, 22, 29, 37, 46,\cdots$

Formula for Central polygonal numbers is $\frac{n(n+1)}{2} + 1$, if $n=1$ or $n$ is prime, we get the new sequence $A$: 2, 4, 7, 16, 29, 67, 92, 154, 191, ... It seems that all primes either is ...
1
vote
1answer
88 views

Combining primes for getting primes?

I was thinking about what would happen when we combine two prime numbers $p$ and $q$ into one number $:pq:$ . Like if $p=5$ and $q=3$ , then $:pq:=53$ . Then if $p=7$ and $q= 11$ then $:pq:=711$ and ...
6
votes
1answer
461 views

Infinite sum of prime reciprocals

Let $\mathbb D$ be the set of all real numbers that can be expressed as a sum of distinct prime reciprocals, i.e. $\mathbb D = \{ d \in \mathbb R \mid d = \sum_{k \in \mathbb K} \frac 1k $ for some ...
1
vote
1answer
68 views

Finding Prime triples with $p_{n} +p_{n+1} −p_{n+2} = 1$

I was just looking at a sequence of primes and suddenly I got this thought that $p_2 +p_3 −p_4 = 1$ since $p_2 = 3, p_3 = 5, p_4 = 7$. Also for $p_3 = 5, p_4 = 7, p_5 = 11$ one has $p_3 +p_4 −p_5 = ...
0
votes
1answer
103 views

Are there composite numbers matching the conditions?

Conditions: n such that $\ Ord_n(2) \mid n-1 $ and $\ Ord_n(2) - 1 = 2^x,n \in >2\mathbb{N}+1,\ x \in \mathbb{Z}_{\geq 0}$. I check up to 1e7 : ...
4
votes
2answers
79 views

Compute $ \lim_{n\to \infty}\prod_{i=1}^n B(p_i^{-2})$

Let $B(x) = \begin{pmatrix} 1 & x \\x & 1 \end{pmatrix}$, and $2=p_1<p_2<\cdots <p_n <\cdots$ primes number. Compute $$\displaystyle \lim_{n\to \infty}\prod_{i=1}^n ...
10
votes
5answers
982 views

Decimal form of irrational numbers

In the decimal form of an irrational number like: $$\pi=3.141592653589\ldots$$ Do we have all the numbers from $0$ to $9$. I verified $\pi$ and all the numbers are there. Is this true in general for ...
4
votes
1answer
132 views

Proof of Infinite Primes in the form $10^{\lceil k \log_{10}(n) \rceil }+n^{k-1}$

Let $k$ be any positive integer then how to prove that the sequence $$Q_k=10^{\lceil k \log_{10}(n) \rceil }+n^{k-1}$$ Contains infinitely many primes? It seems like because if you look at some ...
1
vote
0answers
43 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
3
votes
0answers
157 views

Conjecture on OEIS A167055

OEIS A167055 Numbers n such that $12n + 5$ is prime. $0, 1, 2, 3, 4, 7, 8, 9, 11, 12, 14, 16, 19, 21,...$ are items of OEIS A167055. I conjecture that the set of the sum of every two items of this ...
4
votes
1answer
99 views

Is this sequence monotonically decreasing?

Let $a_n = \frac{p_n - p_{n-1}}{p_n \log p_n}$ where $p_n$ denotes the $n$-th prime. Is this sequence decreasing (or decreasing after some $N$)?
9
votes
3answers
214 views

Does $\sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p}$ converges?

Does $$\sum_{p\in\mathbb P}\frac {( - 1)^{[\sqrt p\,]}}{p}$$ converges ? I know that the following $\sum_{p\in\mathbb P} \frac{1}{p}$ diverges, we can find proofs on Wikepedia Divergence of the ...
0
votes
2answers
52 views

sum of primes: approximate closed form?

Can you find this sum? $$\sum_{\text{primes } p \le n}p.$$ I don't know how to start, let alone do this sum! Thanks for your help. Kind regards.
0
votes
2answers
51 views

Convergence of $\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$

Does this diverge or converge ?? $$\sum_{n=1}^{\infty}(\frac{H_n}{p_n}-\frac{n}{n^n})$$ where $H_n$ is the nth harmonic number, $p_n$ is the nth prime. My impression is that it diverges, but I don't ...
1
vote
1answer
94 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
82 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, ...
3
votes
1answer
191 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 ...
4
votes
2answers
61 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
44 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
136 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 $n$)
41
votes
5answers
4k 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 ...
12
votes
3answers
282 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
65 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} - ...
2
votes
0answers
73 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
121 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
149 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 ...
7
votes
1answer
180 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 ...
20
votes
2answers
391 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
40 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: ...
16
votes
2answers
240 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
145 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
86 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 ...
10
votes
2answers
130 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 ...