Tagged Questions
12
votes
3answers
311 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
74 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?
13
votes
1answer
103 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
0answers
40 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 semiprime ...
4
votes
1answer
113 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?
3
votes
3answers
128 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 ...
6
votes
1answer
56 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
75 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.
0
votes
0answers
51 views
Iterate over combinations ordered by sum
I have a sorted list of a large number of primes. I want to iterate over combinations of fixed size $n$ in increasing order of their sum. Naturally the standard approach for $n=4$:
$$s_0 = \sum(A, ...
1
vote
2answers
26 views
Is there a pattern (or a name and expression for the pattern) of the intervals between all primes?
With the recent interest in Mersenne primes, I got thinking whether there was any mathematical expression for the pattern of intervals (or sequence composed of interval lengths) between ordinary prime ...
5
votes
1answer
113 views
Sequence involving primes of form $n^2 + n+1$
Looking at prime numbers $p_i $ of the form $n^2+n+1$ and the derived expression
$$1 - \prod_{i=1}^{j}\frac{(p_i-1)}{p_i}$$
it seems (I do not claim it and do not see why it should be true) that ...
0
votes
1answer
101 views
Are sequences useful?
Here is a sequence; experimental mathematics :
$2, 2, 2, 11, 11, 254908033,...$ we could define as :
Least primes such that
$((p+1)(nextprime(p)+1))-1$ is prime and
...
2
votes
1answer
57 views
Is there a sequence of primes whose decimal representations are initial segments of each other?
I.e., is there a sequence of primes whose decimal expansions have the following form:
$$a_1,\ a_1a_2,\ a_1a_2a_3,\ a_1a_2a_3a_4, \dots$$
What about with the order of the digits reversed, so each ...
0
votes
1answer
149 views
Evaluating the sum of $\omega(n)$ in an arithmetic progression [closed]
Let $\omega(k)$ count how many distinct prime factors k has,
Then I can prove that for any coprime integers $a,b$
$$\lim_{n\to\infty}\frac{\sum_{k=2}^n\omega(ak+b)}{\sum_{k=2}^n\omega(k)}=1$$
Does ...
2
votes
1answer
117 views
Vonmangoldt sums
The dirichlet series for the Vonmangoldt function, $\Lambda(n)$, which is equal to zero when $n$ is not a prime a power, and $ln(p)$ when it is a prime power say, $n=p^j$, is
...
2
votes
1answer
77 views
Re-writing a series, involving prime numbers
Consider the limit $$\lim_{n \rightarrow \infty} \sum_{k=0}^n \frac{\Lambda(4k+3)}{(4k+3)}-\frac{1}{2}\ln(4n+3)$$
Where $\Lambda(n)$ is the vonmangoldt function, that is equal to zero if n is not a ...
7
votes
1answer
125 views
Convergence of alternating series based on prime numbers
I've been experimenting with some infinite series, and I've been looking at this one,
$$\sum_{k=1}^\infty (-1)^{k+1} {1\over p_k}$$
where $p_k$ is the k-th prime. I've summed up the first 35 terms ...
1
vote
3answers
58 views
The infinite sum $ \sum_{m=2}^\infty \space \frac {1} {p_m \space \log\space m} $
Let $p_n$ denote the $n$th prime , for example $p_1$ = $2$ , $p_2 = 3 $ etc. Then is the sum $$ \sum_{m=2}^\infty \space \frac {1} {p_m \space \log\space m} $$ convergent ?
12
votes
2answers
210 views
Prime one heap Nim
I have been working on an interesting problem my lecturer mentioned recently. Prime Nim is a variant of the Nim game where you have a single pile with an arbitrary number $n\in \Bbb N+\{0\}$ of ...
2
votes
0answers
49 views
Gram's series for integral equation
The prime counting function $ \pi(x) $ satisfies the integral equation
$$ \log\zeta (s)= s\int_{0}^{\infty}dx \frac{ \pi (e^{t})}{e^{st}-1} \tag{0}$$
and it has the solution in terms of Gram's ...
3
votes
1answer
72 views
Convergence of $ \sum_{n=1}^\infty (p_n)^{-n}$
How to determine the convergence of
$$
\sum_{n=1}^\infty (p_n)^{-n},
$$
where $p_n$ is the $n$th prime?
1
vote
1answer
59 views
Series of Mersenne primes
If the 'Lenstra - Pomerance - Wagstaff' conjecture is true, there are infinite Mersenne primes. In this case, if we consider the series:
$$S_N=\sum_{k=1}^N \frac{1 }{M_k}$$
where $M_k$ is $k^{th}$ ...
2
votes
0answers
108 views
Partial summation of a harmonic prime square series (Prime zeta functions)
I am trying to find the following series:
$S=\displaystyle\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}\dfrac{1}{p_ip_j},A\leq p_1 < p_2 < \dots < p_n \leq B, \lbrace A,B\rbrace \in \mathbb{N}$
...
1
vote
2answers
67 views
Does this sequence have this interesting property relating to the prime factorization of the index?
Define a sequence as $a_0 = 0$ and $a_n$ equals the number of divisors of $n$ (including 1 and $n$) that are greater than $a_{n-1}$. This is sequence A152188 in OEIS, by the way.
(For example, the ...
3
votes
2answers
134 views
A series with prime numbers and fractional parts
Considering $p_{n}$ the nth prime number, then compute the limit:
$$\lim_{n\to\infty} \left\{ \dfrac{1}{p_{1}} + \frac{1}{p_{2}}+\cdots+\frac{1}{p_{n}} \right\} - \{\log{\log n } \}$$
where $\{ x ...
4
votes
1answer
234 views
What is the next “Tribonacci-like” pseudoprime?
Given the three roots of $x^3=x^2+x+1$. Then we get the tribonacci-like sequence,
$B_n = x_1^n+x_2^n+x_3^n = 3, 1, 3, 7, 11, 21, 39, 71, 131,\dots$
where $B_n = B_{n-1}+B_{n-2}+B_{n-3}$, and the ...
1
vote
1answer
127 views
Sequence of first differences strictly increasing?
If $ \pi (x) $ := number of primes $ \leq x $, the operation
$T(x_{n+1}) = x_{n+1} - \pi(x_{n+1}) = x_n$
gives a sequence whose elements are those for which repeated application of T gives the ...
1
vote
1answer
80 views
A pattern in distribution of near-primes less than $2^n$
Let $\pi_k(2^n)$ be the number of almost-primes (numbers with k factors including repetitions) less than $2^n$. I noticed that for large values of n and values of k near n, a sequence $\{\pi_k\}$ ...
4
votes
2answers
97 views
Property of a sequence involving near-primes
Let $p_k(m)^2:=$ the square of $m^{th}$ number containing k prime factors, including repetitions.
Empirically for smallish numbers and as a conjecture, it appears that for every m and sufficiently ...
5
votes
1answer
147 views
Sum of alternating reciprocals of logarithm of 2,3,4…
How to determine convergence/divergence of this sum?
$$\sum_{n=2}^\infty \frac{(-1)^n}{\ln(n)}$$
Why cant we conclude that the sum $\sum_{k=2}^\infty (-1)^k\frac{k}{p_k}$, with $p_k$ the $k$-th ...
1
vote
1answer
157 views
Prime reciprocals sum
Let $a_i$ be a sequence of 1's and 2's and $p_i$ the prime numbers.
And let $r=\displaystyle\sum_{i=1}^\infty p_i^{-a_i}$
Can r be rational, and can r be any rational > 1/2 or any real ?
ver.2:
...
3
votes
1answer
150 views
Is $k^2+k+1$ prime for infinitely many values of $k$?
Let's define an infinite sequence of positive integers as :
$a_n=k^2+(2n-1)k+2n-1 $ , where $ k,n \in \mathbf{Z^{+}}$
Suppose that one can prove that this sequence contains infinitely many ...
4
votes
0answers
408 views
Convergent sum with primes
If $f(n)$ is a strictly increasing elementary function from the reals to the reals, and $p(n)$ is the $n$'th prime number. Is there any $f(n)$ such that $\sum_{n=1}^\infty\frac{1}{f(p(n))}$ is ...
7
votes
1answer
137 views
Sum of reciprocals of primes
If $p_i$ is an infinite set of distinct primes such that $c=\sum\frac{1}{p_i} < \infty$, must $c$ be a transcendental number?
6
votes
3answers
143 views
For what $t$ does $\lim\limits_{n \to \infty} \frac{1}{n^t} \sum\limits_{k=1}^n \text{prime}(k)$ converge?
The average of all primes is $$\lim\limits_{n \to \infty} \frac{1}{n} \sum_{k=1}^{n} \text{prime}(k) ,$$ which diverges.
What is the smallest $r$ such that for $t>r$, $$\lim_{n \to \infty} ...
4
votes
1answer
139 views
A finite sum of prime reciprocals
How can you prove that $\sum\limits_k \frac1{p_k}$, where $p_k$ is the $k$-th prime, does not result in an integer?
25
votes
3answers
371 views
Sequence of numbers with prime factorization $pq^2$
I've been considering the sequence of natural numbers with prime factorization $pq^2$, $p\neq q$; it begins 12, 18, 20, 28, 44, 45, ... and is A054753 in OEIS. I have two questions:
What is the ...
5
votes
1answer
462 views
Are the primes found as a subset in this sequence $a_n$?
Below is a introduction that contains some background to my question. The question is found at the bottom.
By calculating the eigenvalues of the matrix defined by the recurrence:
$\displaystyle ...
0
votes
1answer
121 views
Compositions of prime numbers
This question is related to numbers found in the OEIS sequence A191837.
In this sequence, $a(2) = 48 = 5 + 7 + 17 + 19$, where the summands of 48 are all prime numbers that are less than or equal to ...
2
votes
2answers
96 views
Estimating number of crossings for Erastothenes' Sieve
In this paper (2.1) I need to understand the formula for the total number of operations: $$\sum_{i=1}^{\pi(\sqrt n)}\frac{n}{p_i} \approx n\ln \ln n + O(n)$$ On a sidenote, since we're only checking ...
1
vote
1answer
276 views
Algorithm for partitioning n into distinct primes
I am looking for an algorithm that will partition a positive integer into distinct primes. The number of partitions is given by this OEIS sequence: https://oeis.org/A000586
To be more specific, I am ...
0
votes
0answers
93 views
If $p$ is prime > 3,what does $m$ equal to within this expression $\sum_{i=1}^{p-1}1/i = \dfrac{m}{n}$ [duplicate]
Possible Duplicate:
General formula for $\sum_1^{p-1} \frac{1}{x}$, where $p$ is an odd prime
If $p$ is prime > 3,what does $m$ equal to $\sum_{i=1}^{p-1}\dfrac{1}{i} = \dfrac{m}{n}$
I need to ...
4
votes
0answers
260 views
Prime zeta definition, multiplication by zero
Wikipedia has a page about the prime zeta function which is defined as follows:
$$P(s)=\sum_{p\;\text{prime}} \frac1{p^s}$$
I entered this additional definition:
Define a sequence:
...
8
votes
1answer
265 views
Sum of cosines of primes
Let $p_n$ be the nth prime number, $p_1=2,p_2=3,p_3=5,\ldots$
How to prove this series converges/diverges?
$$\sum_{n=1}^\infty \cos{p_n}$$
5
votes
1answer
122 views
trajectories in ListPlot[the maximal prime factor of average of twin prime pair]
another sequence on twin primes The maximal prime factor of average of twin prime pair:
n = 100000;
averageList = Select[Prime[Range[n]], PrimeQ[# + 2] &] + 1;
mpfList = FactorInteger[#][[-1, 1]] ...
2
votes
2answers
72 views
numbers interference
a sequence on twin primes In the diagram, How do the stripes come from? Can the prime numbers also interfere like light and wave? When zoom in or zoom out the diagram in Mathematica, the stripes are ...
