1
vote
0answers
68 views

Successive ratios of a sequence, is this limit true?

The natural numbers $1,2,3,4,5....$ can be calculated as the row sums of the triangle $T(n,k)$ equal to $1$ if $n \geq k$ and $0$ otherwise: $$\displaystyle T = \left(\begin{matrix} ...
6
votes
2answers
232 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
2
votes
1answer
127 views

A closed form for the series $\sum_{n=1}^{\infty} \frac{H_n^2-(\gamma + \ln n)^2}{n}$

I have found a closed form for the following new series involving non-linear harmonic numbers. Proposition. $$\sum_{n=1}^{\infty} \dfrac{H_n^2-(\gamma + \ln n)^2}{n} = ...
3
votes
1answer
108 views

Golden ratio, $n$-bonacci numbers, and radicals of the form $\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\sqrt[n]{\frac{1}{n-1}+\cdots}}}$

The following infinite nested radical $$\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$$ is known to converge to $\phi=\displaystyle\frac{\sqrt{5}+1}{2}$. It is also known that the similar infinite ...
1
vote
0answers
62 views

Can there be a power series with interval of convergence $[k, \infty)$?

My answer : NO Because Interval of convergence is of the form $(a-R, a+R)$ Where $a$ is centre of convergence. If there exists a power series with Interval of convergence $[k, \infty)$ $ $ We ...
9
votes
2answers
136 views

How to prove that the number $1+4a_{n}a_{n+1}$ is a perfect square.

A sequence of integer $\{a_{n}\}$ is given by the conditions $a_{1}=1, a_{2}=12,a_{3}=20$,and $$a_{n+3}=2a_{n+2}+2a_{n+1}-a_{n}$$ show that for every postive integer $n$, the number ...
8
votes
0answers
64 views

A sequence that avoids both arithmetic and geometric progressions

Sequences that avoid arithmetic progressions have been studied, e.g., "Sequences Containing No 3-Term Arithmetic Progressions," Janusz Dybizbański, 2012, journal link. I started to explore sequences ...
0
votes
1answer
38 views

a finite induction question from burton's elementary number theory

this question comes from burton's elementary number theory, 4th edition. question 3 in 1.1 says to use the second principle of finite induction to establish that $$for\ all\ n\ge1,\ ^{(a)}\ ...
2
votes
0answers
81 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 ...
26
votes
4answers
493 views

Geometric intuition for $\pi /4 = 1 - 1/3 + 1/5 - \cdots$?

Following reading this great post : What are some interesting cases of $\pi$ appearing in situations that are not / do not seem geometric?, Vadim's answer reminded me of something an analysis ...
2
votes
0answers
86 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
2
votes
0answers
67 views

Unexplanied pattern from increasing rational sequences

I've stummbled upon some strange pattern when working with series of rational numbers. If anyone could shed light on this phenomenon I will be most greatful. Backgroud: Working in an integer lattice ...
1
vote
0answers
20 views

A Variation on the Coin Problem

Suppose I have a sequence $a_n$, whose entries are the ordered elements of $S_{x,y}$: $S_{x, y}= \{ z \mid \left( z=n_1x+n_2y \right) \wedge \left( n_1, n_2 \in \mathbb{N}_1 \right) \wedge \left( ...
0
votes
0answers
48 views

A general question on positive integer sequence of a certain formula

Let $A=\{\ $ a certain polynomial | all variables$\ \in\mathbb N\ \},\ A\ \subseteq\ \mathbb Z^+,\ $ such as $A = \{2n−1\ |\ n\in\mathbb N\}$. Let $B=\mathbb Z_{\ge 0}-A$. Let $C$ be the set that ...
1
vote
1answer
72 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 ...
1
vote
1answer
55 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 = ...
3
votes
1answer
41 views

Infinite Perfect power of numbers in a certain form

A question I found very interesting , which I found written on a blackboard while visiting a near by community science center is as follows. Prove that there exist infinitely many $m,n,k$ for ...
18
votes
0answers
209 views
+50

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
1
vote
0answers
22 views

number set with non-adjacent digits

I'm not sure such thing exists, or even if I'm asking for a valid thing, but I'll do my best describing it. At least the thing I want is similar to Gray code, so should be valid. So, I want to know ...
2
votes
3answers
316 views

Proof of irrationality of a series

Given the series $$S=\sum_{k=1}^N\frac{1}{k^q}$$ $(q\gt0),(N\in\mathbb{N},N\ge1)$ is $S$ irrational for every choice of $q$ and $N$? Thanks.
1
vote
0answers
60 views

How to generilize the the following summation.

While searching for a summation formula I come accross the following equation on wikipedia Equation $$\sum\limits_{k=1}^{n}{k^m z^k}=\left(z\frac{d}{dz}\right)^m\frac{z-z^{n+1}}{1-z}$$ So I tried to ...
0
votes
0answers
42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
8
votes
1answer
190 views

Help me ID this weird $\pi$ formula

I remembered, and managed to find, still gathering dust in a forgotten corner of the Internet, an old QuickBASIC program which, with a trick, can rapidly sum up a HUGE amount of terms of the famous ...
1
vote
1answer
36 views

Does sequences related to function for $lcm(1,2,3 \cdots n)$ exists?

This just came out of curiosity let $$L(n)=lcm(1,2,3 \cdots n)$$ and I know that we can write this with the help of some product involving primes and all . But what I am interested is in Does ...
2
votes
0answers
49 views

Numbers for $k^{th}$ power of sum of digits equal the sum of digits of $k^{th}$ power

Let us denote $D_*(n)$ to be the sum of digits of $n$. Then we can find numbers satisfying the formula $$D_*(n^2)=D^2_*(n)$$ Like : $11^2 =121 ,\; \;(1+1)^2= 1+2+1$ , other like $12^2=144, \; \; ...
0
votes
1answer
58 views

Question about the convergence of an infinite series in all of $\mathbb{C}$.

Does the following infinite series: $$G(s):=\displaystyle \sum^{\infty}_{n=1} \frac{(-1)^{n+1}}{n^s + \frac{1}{n^s}}$$ converge for all $s \in \mathbb{C}$ (especially in the critical strip; since ...
2
votes
1answer
77 views

On combining $n$ and $n^2$ into one number

Consider the sequence $T_n$ formed by combining $n$ and $n^2$ into one number. ie. (A053061) $$T_n=\{11,24,39,416,525,636,749 \cdots\}$$ It is easy to see $$T_n= 10^{\lceil 2 \log_{10}(n) \rceil } n+ ...
2
votes
0answers
59 views

Formula for sequence of integers

I am trying to compute the Taylor series of $f(x) = \sqrt{-\ln(x)}$. I compute the derivatives of $f(x)$ and evaluate them in the point $x=1/e$. The resulting expressions have the following ...
0
votes
1answer
59 views

A complex series with exponentials

I have tried to solve this type of series : $$\sum \frac{e^{i\, u(n)}}{v(n)} $$ For some $u,v$ an Abel Transform allow to find convergence, but for $u(n)=n^2$ and $v(n)=n$ I can't find an argument. ...
0
votes
1answer
87 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 : ...
0
votes
1answer
73 views

Summing up numbers from the continued fraction of $e ^ \pi$ and $\pi ^e$

I don't remember it well ,but it was around 5-6 years ago , I was 8 and I had found this new interest - continued fractions .I used to play with their terms sum them up and thought of getting ...
7
votes
1answer
175 views

The closed form of $\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor\sqrt{3n}\rfloor^2}-\frac{1}{3n}\right)$

I need some ideas to exploit for finding the closed form of $$\sum_{n=1}^{\infty} \left(\frac{1}{\lfloor\sqrt{3n}\rfloor^2}-\frac{1}{3n}\right)$$
10
votes
5answers
904 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 ...
7
votes
1answer
123 views

How prove $a_{n}=[\sqrt{2}n]+[\sqrt{5}n]$ Contains infinitely even numbers.

let sequence $$a_{n}=[\sqrt{2}n]+[\sqrt{5}n]$$ where $[x]$ is the largest integer not greater than $x$ show that $\{a_{n}\}$ Contains infinitely even numbers. also I guess ...
2
votes
0answers
60 views

All those unit fractions add to 1?

Consider $$S(n)=\{x \mid x=(a_1 ,a_2,a_3 \cdots a_n) \text{ where } \sum_{r=1}^{n}\frac{1}{a_r} =1 \}$$ Now let $|S(n)|$ denote the cardinaly (order) of set $S(n)$. Thus: $S(1)= \{(1)\} \implies ...
4
votes
1answer
120 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 ...
7
votes
2answers
115 views

On the numbers divisible by all the Integers not exceeding their $r^{th}$ roots.

Consider the set of all numbers which are divisible by all natural numbers not exceeding their square root, and denote this set by $S_2=\{1,2,3,4,6,8,12,24\}$ (Here the subscript indicates that we're ...
1
vote
0answers
41 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
152 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 ...
3
votes
2answers
93 views

why generating function $A(z) = 1 + z + z^2 + \cdots$ can be denoted as $\frac{1}{1-z}$

It is easy to see that $1 + z + z^2 + \cdots$ is equal to $\frac{1}{1-z}$ when $1 > z > 0$ and for $z >= 1$, they are not equivalent. So I have thought $\frac{1}{1-z}$ is just a short for the ...
9
votes
3answers
205 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 ...
4
votes
1answer
87 views

Euler's Refutation of Fermat's Conjecture

Fermat postulated that all numbers of the form $$2^{2^n}+1$$ are prime (where n = any integer). Then Euler came along with a rather ingenious proof that this was not, in fact the case. I came across ...
8
votes
3answers
206 views

Finding 1000th 5-smooth number

A number is 5-smooth if its only prime factors are $2,3$ or $5$. Example: $$1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, \dots$$ Interesting thing is that as they become larger and larger, they are ...
1
vote
3answers
144 views

if $ a_{n+1}=n^2+3\sqrt{\frac{3a_{n}-a_{n-1}}{2}}$ this sequence have infinite number of “good” term

Question: Given a sequence $\{a_{n}\}$, call a term $a_{k}$ "good", if there exist $a_{m},a_{n}$, such that $$a_{k}=a_{m}a_{n}$$($a_{m}$ and $a_{n}$ are allowed to be equal.) Otherwise it is ...
1
vote
3answers
50 views

About the calculation of decimal digits of series up to the nth digit

Considering that we don't know any of the digits of some number defined as the limit up to infinity of a sum, I want to know how many terms do I have to sum to get the correct decimal representation, ...
0
votes
0answers
27 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} + ...
1
vote
0answers
34 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
989 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
1answer
25 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
47 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 ...