0
votes
0answers
14 views

Question regarding the sum of the reciprocal of the values in Sylvester's sequence

The unit fractions formed by the reciprocals of the values in Sylvester's sequence generate an infinite series: $\sum_{i=0}^{\infty} \frac1{s_i} = \frac12 + \frac13 + \frac17 + \frac1{43} + ...
0
votes
0answers
29 views

Determing sequence from its Dirichlet series

Suppose I know the Dirchlet series $$\sum_{n=1}^{\infty} \frac{f(n)}{n^s} = \frac{\zeta(s)}{\zeta(3s)},$$ where $\zeta(s)$ is the usual Riemann zeta function. My question is - is there a way to ...
10
votes
3answers
972 views

Is this already an equation/law that has been found?

So I was messing around with some numbers today and I have found a way to quickly add summations (probably not the first one to discover it but...) this only works when you start at 1 (i.e. ...
0
votes
0answers
7 views

Particular type of input to gather information on certain types of encryption schemes

Consider a simple case in which information is sent is considered as it's binary equivalent and then those numbers are considered as base 10 and used as inputs to an equation over a number field of ...
0
votes
1answer
24 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
1
vote
3answers
43 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
0
votes
0answers
26 views

Logarithmic derivative of Riemann zeta, is this derivation correct?

Let matrix $T_2$ be defined below as the Dirichlet inverse of the Euler totient function as a function of the Greatest Common Divisor (GCD) of row index $n$ and column index $k$; $$T_2(n,k) = ...
6
votes
1answer
108 views

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer

Find all natural numbers such that $\sum_{k=1}^{n} \frac{n^k}{k!}$ is an integer. I've tried to bring all fractions under commmon denominator and it didn't helped me much. With guessing I find out ...
0
votes
2answers
31 views

sequence $a_n = \lceil \sqrt{2}n \rceil $

I was trying to prove $\lceil \sqrt{2}n \rceil + \lceil \sqrt{2}m \rceil \geq \lceil \sqrt{2}(n+m) \rceil$ where $m,n\in \mathbb{z}$ Direct proof I tried but could not figure out. I tried fixing m ...
2
votes
0answers
268 views

First disagreement in PROUHET THUE MORSE exponentially big?

Let two sequences of integers be $a_1, \cdots, a_n$ and $b_1, \cdots, b_n$ such that with $a_i \in \{1, \cdots n\}$ and $b_i \in \{1, \cdots, n\}$. Let $k$ be the min integer such that $\sum_{i=1}^n ...
3
votes
1answer
35 views

Lower bound on certain exponential sums and expressions related to them

Let $$G(\alpha, x) = \sum_{n\le x}e(\alpha n^2)$$ Clearly, $r_k(n)$, the number of representations of a number as the sum of $k$ squares is given by the following expression: $$r_k(n) = \int_0^1 ...
1
vote
0answers
57 views

Irrational numbers and series

Let $$f(x) = \prod_{n = 0}^\infty \left(1 + \frac{x}{2^n}\right)$$ According to an exercise in a packet of problems in elementary number theory, this function and all its derivatives are irrational ...
1
vote
2answers
61 views

Proving the sum of the harmonic series up to $p-1$ is divisible by $p$

Wolstenholmes theorem says that $1 + \frac{1}{2} + ... + \frac{1}{p-1} \equiv 0 \bmod p$. I dont quite get the proof, but I was wondering if the argument here is valid for the narrow case that it is ...
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
127 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 $n$)
0
votes
0answers
35 views

if $F_{1}=F_{2}=1,F_{n+2}=F_{n+1}+F_{n}$ ,then there exist $F_{i}$,such $p|F_{i}$ [duplicate]

Let sequence $\{F_{n}\}$ such $$F_{1}=F_{2}=1,F_{n+2}=F_{n+1}+F_{n}$$ let $p$ is prime number,show that:there is exsit $F_{i}$ such $$p|F_{i},1\le i\le p+1$$ my idea: if $$p=2$$,then then ...
2
votes
0answers
102 views

Equality between an infinite product and an infinite series. How can I reconcile both?

Maybe a trivial question, but how could I reconcile the following equation: $$\displaystyle \prod_{n=2}^\infty \left(\frac{1}{1-\frac{1}{n^2}}\right)^{(-1)^n}=\sum_{n=1}^\infty \left(\frac{1}{(2\,n ...
0
votes
1answer
62 views

Why this difference of 25?

The numbers 4962, 29922, 179862, and 1389858 share a curious property and I am wondering if someone can enlighten me.  The largest prime factor of each number differs, by 25, from the remainder of the ...
38
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 ...
1
vote
1answer
52 views

Check if a series can be made or not

Given an infinite geometric sequence $S=\{1, \frac{1}{2}, \frac{1}{4}, \frac{1}{8}, \dots \}$. I need to tell whether for given fraction $p⁄q$, we can select an infinite geometric sequence $R$, such ...
0
votes
0answers
33 views

Relating sequence of integers with decimal representation

Let a sequence of integers be defined by a rule/formula,if this sequence is supposed to be a decimal representation of a number such as: $1)$ $a_{n+1}=a_n+1$ (given $a_0=0$) ...
2
votes
2answers
72 views

Traversing integer spiral in a different direction

Arrange the positive integers in a counter-clockwise spiral (often referred to as Ulam's spiral), beginning with 1 and starting out east, north: ...
0
votes
0answers
70 views

Proof of a striking identity of Tito Piezas III

In the q series blog of Tito Piezas here . He gives a very striking relation I am wondering on how to prove that ?
3
votes
0answers
50 views

Which series of numbers effectively translates the factorial to the exponential function?

We have the relation of the Bernoulli numbers $$B_{2n} = (-1)^{n+1}\frac {2(2n)!} {(2\pi)^{2n}} \left(1+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\cdots\;\right).$$ For $n>1$, the right hand sum ...
1
vote
0answers
69 views

All about a failed conjecture.

Some months ago I made the following conjecture - Let $d(n)$ denote the number of divisors of $n $. Then let $N$ be a number such that $d(N)$ divides $N$ . Also let $I= \frac{N}{d(N)}$ which is ...
4
votes
1answer
72 views

How prove this sequence $a_{n}=a_{n-1}+\frac{1}{a_{n-1}}$,then $a_{p}+a_{q}\notin Z$

let sequence $\{a_{n}\}$ such $$a_{1}=\dfrac{1}{2},a_{n}=a_{n-1}+\dfrac{1}{a_{n-1}}$$ show that: for any $(p.q)\neq (1,2),p\neq q$,such $a_{p}+a_{q}\notin Z$ My idea: since ...
6
votes
1answer
159 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 ...
3
votes
1answer
91 views

The relationship between Golbach's Conjecture and the Riemann Hypothesis

My question pertains to two famous groups of related conjectures: Goldbach's Conjecture (GC); Goldbach's Weak Conjecture (GWC); The Riemann Hypothesis (RH); The Generalized Riemann Hypothesis (GRH). ...
2
votes
3answers
70 views

How find this positive integer numbers $m$,such $a_{n}=2a_{n-1}+a_{n-2}$,if $2^{2011}|a_{m}$

let sequence $$a_{0}=0,a_{1}=1,a_{n}=2a_{n-1}+a_{n-2},(n\ge 2)$$ Find all positive integer numbers $m$ ,such $2^{2011}|a_{m}$ My try:since $a_{0}=0,a_{1}=1$,then ...
1
vote
2answers
39 views

How can we make any integer m>11 using 3's and 5's only?? [duplicate]

Is there any general solution of this? using 2 integers, what is the minimum number formed after which we can make any number using those 2 integers? so it says 3a + 5b = m now we know 4, 7 do not ...
5
votes
1answer
142 views

Sequence where the sum of digits of all numbers is 7

BdMO 2014 We define a sequence starting with $a_1=7,a_2=16,\ldots,\,$ such that the sum of digits of all numbers of the sequence is $7$ and if $m>n$,then $a_m>a_n$ i.e. all such numbers are ...
1
vote
1answer
61 views

Eliminating numbers from the sequence $1,2,3,4,5,6,7…400$

BdMO 2014 Let us take the sequence $1,2,3,4,5,6,7....400$ .We are going to remove numbers from the sequence such that the sum of any 2 numbers of the remaining sequence is not divisible by 7.What ...
7
votes
1answer
151 views

Intuitively, why is the Euler-Mascheroni constant near sqrt(1/3)?

Questions that ask for "intuitive" reasons are admittedly subjective, but I suspect some people will find this interesting. Some time ago, I was struck by the coincidence that the Euler-Mascheroni ...
4
votes
0answers
54 views

Closest Pair between 2 sets of powers

Assume $a$ and $b$ are natural numbers, $A=\{a,a^2,a^3,\cdots\}$ and $B=\{b,b^2,b^3,\cdots\}$, find $\min\left|a_i-b_j\right|$ where $a_i\in A$ and $b_j\in B$. For example, if $a=3$, $b=10$, then ...
0
votes
2answers
56 views

Series equivalent to differences of inverse primes?

Would someone be kind enough to correct my error here? $S=1+2+3+4+\cdots$ Now starting with the 1st prime number, regroup the sum: $S=(1+3+5+7+\cdots) + (2+4+6+\cdots)$ $S=(1+3+5+7+\cdots) + ...
0
votes
1answer
86 views

a Problem about Sequence [duplicate]

Let $a_1$ be an integer. Then we assume $$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$ Now we prove that for any ...
2
votes
0answers
96 views

A connection between a sequence from the Collatz conjecture and a sequence of densities from $\zeta(k)-1$?

Just for grins, I created lists of first-entries of finite sequences of rank $r$ for the Syracuse problem (Collatz conjecture using only odd numbers) and found these sequences on OEIS. My sequences, ...
0
votes
0answers
91 views

An identity about Dirichlet $\eta$ Function

We know the Dirichlet $\eta$-function is defined as the analytic continuation of $$\eta(s) = \sum_{i=1}^\infty \frac{(-1)^{n-1}}{n^s} \quad \Re(s)>0$$ I find an identity for the values of this ...
2
votes
0answers
46 views

Distribution of Omega values modulo m

Define $\Omega(2^{a_1}3^{a_2}...p_k^{a_k})=a_1+...+a_k$ . I am interested in the density of the values of Omega mod m. If we define the set $S=(x:\Omega(x)\equiv k \text{ mod m})$, I would like to ...
1
vote
0answers
59 views

Is this a bounded sequence ? (about continued fraction)

Represent $\sqrt{2}$ in the form $$\sqrt{2}=1+\frac{8}{A_1+\displaystyle\frac{8}{A_2+\displaystyle\frac{8}{A_3+\ddots}}},$$ where $A_n$ is a positive integer and $A_n \geq 8$ for all $n$. So we have ...
2
votes
1answer
77 views

Natural Density and Logarithmic Density

Natural density of a set $S = (a_1,a_2,...) $ is defined (assuming it exists) as $$\lim_{n \to \infty}\frac{1}{n}\sum\limits_{k\in S, k\le n}1 $$ The logarithmic density of the same set is defined as ...
14
votes
2answers
178 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
2answers
180 views

sum of infinite roots? [duplicate]

What is the sum of this infinite series of roots: $$\sqrt{1+\sqrt{2+\sqrt{3+\sqrt{4 + \cdots+\sqrt{\infty}}}}}$$ This is an interesting expression because the increase created by the addition of the ...
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 ...
0
votes
0answers
27 views

About a specific mathematical series which is a power of the exponential function

My professor wrote the below exponential function just out of the box when he suggested a kernal for a 1D domain. $f(x) = e^{-\Big(a_1x_1+ \dfrac{1}{2} a_2 x_2^2 + \dfrac{1}{3} a_3 x_3^3 + .... ...
2
votes
0answers
41 views

Infinite series with only two zeros at $\Re(s)=\frac12$. Why is that the case?

I was experimenting with the following series: $$\displaystyle k(s)=\sum_{n=1}^\infty \frac{1}{n^{\ln (\frac{n}{s})}}$$ I believe this is an entire function without any poles in $\mathbb{C}$ (note ...
2
votes
2answers
82 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 ...
3
votes
1answer
47 views

A function whose real part only converges (to zero) for $\Re(s)=\frac12$?

As a follow up on this question, I like to conjecture that for $s \in \mathbb{C}, n \in\mathbb{N} /0$, the real part of: $$\displaystyle f(s):=\lim_{n \to +\infty} \left( \frac{s-1-2\,n}{2\,(s-1)\, ...
9
votes
2answers
121 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 ...
8
votes
3answers
321 views

On the “Look-and-Say” sequence and Conway's constant

The look-and-say sequence starting with $S_1=1$ is, $$S_n = 1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211,\dots$$ If $L_n$ is the number of digits of the $n$th term then, ...