1
vote
1answer
26 views

Simplify a power series

I am studying bernouli numbers and I'm having trouble condensing a power series. In particular, I'm studying the equation $$b(x)^2=(1-x)b(x)-xb'(x)$$ where ...
0
votes
0answers
35 views

counting the good numbers

We need to calculate Good Numbers in range from $A$ to $B$ (Both inclusive). A number $N$ is said to be a good Number if it satisfy following conditions : If we extract every $2$-digit number of $N$ ...
4
votes
0answers
51 views

The “Largeness” of Subsets of the Natural Numbers

Recently, I was thinking about the Erdős conjecture on arithmetic progressions, which says that, if, for a set $A\subseteq\mathbb{N}$, the sum $\sum_{a\in A}\frac{1}a$ diverges (i.e. "A is large"), ...
0
votes
1answer
21 views

show existence of subsequence $\{a_{i_b}\}_b^{n+1}$

Suppose $\{a_n\}_{n=1}^{m^2+1}$ is a strictly increasing sequence of $n^2+1$ positive integers, show that there exist a subsequence $\{a_{i_b}\}_b^{n+1}$ of length $n+1$ such that $a_{i_k}$ is ...
0
votes
1answer
20 views

What are the total number of ordered permutations of a list of elements (possibly repeated)?

This question is a part of a TopCoder problem. I am learning algorithms, and just got stuck at this (not homework). Suppose we have a set $A$ of integer elements, such that $n(A) = a$ (number of ...
0
votes
1answer
27 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 ...
1
vote
0answers
18 views

Sequence terms being divisible

Here's a question I would like hints for: The sequence ${x_n}$ is defined by $x_{n+2}=6x_{n+1}-9x_{n}$ for $n \geq 0$ where $x_0=3$ and $x_1=18$. What is the smallest $k$ such that $x_k$ is ...
0
votes
0answers
23 views

Unique sum between elements of a numerical set

I require a set of numerical elements on wich the sum of some of these elements is unique to the set, that it's to say, no other combination in the sum of elements will result the same outcome. ...
0
votes
0answers
28 views

Given a positive integer $k$, find the integer part of $n^2 /k$ for $n\ge 1$, and a related question.

For a given positive integer $k,$ I am looking for possible answers / literature about the sequence $(a_n)=([\frac{n^2}{k}])_{n=1}^\infty$, where $[x]=$the integer part of $x.$ This question is ...
0
votes
1answer
43 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 ...
-2
votes
2answers
138 views

How to find sum of 4th power of n numbers mod m [closed]

How can i calculate $1^{4} + 2^{4} + 3^{4} + 4^{4} .....+n^{4} \pmod m$ where $1 \le m \le 10^5$ and $1\le n \le 10^{20}$. I can't use the formula here because it will Overflow the limit of long long ...
3
votes
1answer
98 views

golden ratio from new formula? perhaps from theory of modular units?

Please consider the following infinite product series which I found by pure happenstance: $$\frac{1+\sqrt{5}}{2}= e^{\pi/6} \prod_{k=1}^\infty \frac{1+e^{-5(2k-1)\pi}}{1+e^{-(2k-1)\pi}}$$ My ...
4
votes
7answers
108 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 ...
8
votes
2answers
518 views

Is 641 the Smallest Factor of any Composite Fermat Number?

Consider the sequence $a_n = 2^{2^n}+1$ of so-called Fermat numbers. It's well known that $a_5$ isn't prime ($a_5 = 641 \cdot 6700417$, this is due to Euler). What I want to know about this sequence ...
5
votes
3answers
79 views

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$?

How would you show that the Riemann Zeta function, $\zeta(s) < 0$ for $s \in (0,1)$? So far I have that along the critical strip \begin{align} \zeta(s) &= ...
1
vote
0answers
55 views

Find a closed form for the constant term

In a previous question, an asymptotic expansion was provided for the weighted divisor summatory function $\displaystyle \frac {d(n)}{n}$: $$\sum_{n\leq ...
2
votes
0answers
139 views

Closed formula for the numbers of the form $\sqrt{1+\sqrt{4+\sqrt{9}}}$

how can i find the formula for the nth term of this series? SQ = square root $\sqrt{1} = 1$ $\sqrt{1 +\sqrt{4}} = \sqrt{3}$ $\sqrt{1 +\sqrt{4+\sqrt{9}}} \approx 1.909385061$ $\sqrt{1 ...
0
votes
0answers
18 views

Identifying or bounding the zeros of the composition of two generating functions

Given two generating functions $$ G(a_n;x)=\sum_{n=0}^\infty a_nx^n \quad\text{ and }\quad H(b_n;x)=\sum_{n=0}^\infty b_nx^n, $$ what techniques are available for locating, or finding bounds on, the ...
-1
votes
1answer
52 views

looking at the alphabet ,the letters are numbered 1-26 ,

looking at the alphabet ,the letters are numbered 1-26 , such that 1 =one=15+14+5=34 (O=15, N=14, E =5 ) 2=two=20+23+15=58 (T=20, W=23, 0=15) 3=three =56 4=four=60 ...
4
votes
6answers
645 views

Sum of an unorthodox infinite series

$ \frac{1}{2^1}+\frac{3}{2^2}+\frac{5}{2^3}+\frac{7}{2^4}+\cdots $ This is a pretty unorthodox problem, and I'm not quite sure how to simplify it. Could I get a solution? Thanks.
8
votes
2answers
169 views

Convergence of the sequence $\frac{1}{e^k \sin{k}}$

Does the sequence $\frac{1}{e^k \sin{k}}$ converge? If $\sin{k}$ acts as a random variable (taking on values in $(-1, 1)$), then it seems like we should be able to prove that the sequence converges ...
4
votes
1answer
267 views

Prove or disprove that there exists a unique positive integer sequence $\{a_{n}\}$ satisfying a condition

Question: Prove or disprove: there exists a unique positive integer sequence $\{a_{n}\}$ satisfying the following condition: $\forall m\in N^{+}$, there exists a unique integer sequence ...
3
votes
0answers
37 views

Generalization of small set/large set

A small set is a subset of the positive integers, such that the infinite sum of the reciprocals of the members of the set converges. Conversely, the sum of the reciprocals of a large set diverges. ...
0
votes
1answer
41 views

Convergence of $\sum_{n=1}^{\infty} n$ and integral test [duplicate]

I have read online, that one can show that $\sum_{n=1}^{\infty}n = -\frac{1}{12}$. But isn't this a Riemann Series of the form $\sum_{n=1}^{\infty} \frac{1}{n^p}$, where $p=-1$. And if so, can't ...
1
vote
0answers
65 views

sum of reciprocal power-1

I found this in my old notebook $$\sum_{n \text{ perfect power}} {\frac{1}{n-1}} = 1$$ and this was my "proof" $$ \begin{align} \frac{1}{1}+\frac{1}{2}+\cdots ...
4
votes
1answer
71 views

Limit of differences of truncated series and integrals give Euler-gamma, zeta and logs. Why?

In the MSE-question in a comment to an naswer Michael Hardy brought up the following well known limit- expression for the Euler-gamma $$ \lim_{n \to \infty} \left(\sum_{k=1}^n \frac 1k\right) - ...
1
vote
1answer
34 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: $\displaystyle\forall{0}\leq{i}\leq{m}:f(i)=P_i$ ...
6
votes
1answer
160 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 ...
0
votes
1answer
122 views

Calculating of the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$? [closed]

How can i calculate the sum $\sum_{i=1}^{n} \large{\frac{i}{i+1}}$?
0
votes
0answers
62 views

Proof of convergence of Kaprekar's Constant

I've tried googling this one a bit but nothing seems to come up, even though its considered to be a well known fact. Why does the kaprekar process of taking a 4 digit number: L, generating L' and L'' ...
3
votes
0answers
158 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} ...
9
votes
2answers
333 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 ...
5
votes
1answer
178 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
121 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
63 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
139 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 ...
9
votes
0answers
79 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
48 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
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 ...
26
votes
4answers
515 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 ...
3
votes
0answers
165 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
72 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
22 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
49 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
86 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
65 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
46 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 ...
27
votes
1answer
559 views

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
27 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
320 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.