2
votes
2answers
86 views

A transcendental number from the diophantine equation $x+2y+3z=n$

Let $\displaystyle n=1,2,3,\cdots.$ We denote by $D_n$ the number of non-negative integer solutions of the diophantine equation $$x+2y+3z=n$$ Prove that $$ \sum_{n=0}^{\infty} ...
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)}\ ...
5
votes
1answer
62 views

How many decimal representations are possible for the number 1

I know that there at least two $0.\overline{9}$ and 1 Is there a neat and more concrete way to understand this problem.
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 ...
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 $A = \{2nāˆ’1\ |\ n\in\mathbb N\}$. Let $B=\mathbb Z_{\ge 0}-A$. Let $C$ be the set that ...
2
votes
0answers
41 views

Proving a quantity negative.

For $j\in{1,2,3}$ let $x_j,y_j \in R$ be nonzero and let $v_j=x_j+y_j$. Suppose that following holds: $$x_1x_2x_3=āˆ’y_1y_2y_3 \quad \text{and} \quad x^2_1+x^2_2+x^2_3=y^2_1+y^2_2+y^2_3$$ nd that ...
1
vote
0answers
61 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
31 views

Frogs on lotus trees [duplicate]

$n(>1)$ lotus leaves are arranged in a circle. A frog jumps from a particular leaf by the following rule: It always moves counter clockwise. From starting point it skips one leaf and jumps to the ...
6
votes
3answers
940 views

To prove the sequence 11,111,1111,… does not contain a perfect square number. [duplicate]

I have to prove that the sequence $\{11,111,1111, \dots \} $ doesn't contain any perfect square numbers. I can realize it but I am unable to prove it. Please help.
1
vote
2answers
77 views

Find $\sum\frac{a(n)}{n(n+1)}$, where $a(n)$ — number of 1's in binary expansion of n. [duplicate]

Let $a(n)$ is a number of 1's in binary expansion of n, find the sum $$ \sum\limits_{n=1}^{\infty}\frac{a(n)}{n(n+1)}. $$
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 ...
14
votes
1answer
116 views

How to sum this infinite series

How to sum this series: $$\frac{1}{1}+\frac{1}{11}+\frac{1}{111}+\frac{1}{1111}+\cdots$$ My attempt: Multiply and divide the series by $9$ ...
1
vote
0answers
18 views

Commuting limits in relating the harmonic series to coprimality densities

Let $f(b, q)$ be the reciprocal proportion of naturals less than $b$ which are not divisible by any prime less than $q$. Note that $\displaystyle \lim_{q \to \infty} f(b, q) = b$, while $\displaystyle ...
1
vote
1answer
35 views

Help with this hard recurrence relation question

Please help with this. Suppose $\{a_n\}$ satisfies $$a_n=(n+1)a_{n-1}-(n-2)a_{n-2}-(n-5)a_{n-3}+(n-3)a_{n-4},$$ and $a_0=a_1=1,a_2=a_3=0$. Please sort out the general form of $a_n$. I guess $a_n$ ...
4
votes
1answer
55 views

Number of occurances of a number $n$ in $\lfloor \sqrt{0} \rfloor, \lfloor \sqrt{1} \rfloor, \lfloor \sqrt{2} \rfloor, \dots$

Recently, while learning Python (the programming language), I started playing around to generate the following sequence: $$\lfloor \sqrt{0} \rfloor, \lfloor \sqrt{1} \rfloor, \lfloor \sqrt{2} ...
3
votes
1answer
60 views

Number of palindromic numbers less than a power of $10$

I noticed that every $10^{n}$ there is a certain number of palindromic numbers that I collected in this sequence: $$S=\{a_n,a_{n+1},a_{n+2}...\}=\{10,9,90,90,900,900...\}$$ where every number $a_n$ is ...
0
votes
1answer
40 views

Is there name to the following sequence: $c_n = c_1c_2…c_{n-1} + 1$

I just saw the sequence $c_n = c_0c_1c_2...c_{n-1} + 1$ and is thinking whether sequence $(c_n)$ has some name. Add: What if $c_0 \neq 2$?
0
votes
2answers
71 views

summation of ceil and floor function

I need a closed solution or a faster algorithm for calculating $$ \sum_{k=1}^{n-1} \left\lceil \frac{n}{k}-1 \right\rceil $$ and $$ \sum_{k=1}^{n-1} \left\lfloor \frac{n}{k} \right\rfloor $$ where $ ...
0
votes
0answers
31 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
122 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 ...
1
vote
0answers
60 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 ...
5
votes
1answer
166 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
65 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 ...
9
votes
2answers
155 views

Has anyone noticed this pattern?

I've been messing around a bit and I noticed a curious pattern when it comes to progressions of powers. Let's take the progression of consecutive integers: $1,2,3,4,5,6,7,...$ Obviously it's an ...
1
vote
5answers
112 views

Sum of elements in the nth set of the sequence of sets of squares $\{1\}$, $\{4,9\}$, $\{16,25,36\}$, …

Let $S_n$ denote the sum of the elements in the $n^{th}$ set of the sequence of sets of squares: $\{1\}$, $\{4,9\}$, $\{16,25,36\}$, $\{49,64,81,100\}$,.... i.e. $S_1 = 1$, $S_2 = 13$, ... How do you ...
2
votes
0answers
49 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 ...
3
votes
1answer
223 views

Find the $n^{\rm th}$ digit in the sequence $123456789101112\dots$

Basically, the question asks us to find the nth digit in the following sequence: $$12345678910111213\dots9899100101\dots$$ where the 10th digit is $1$, the 11th digit is $0$, etc. EDIT: Here are my ...
1
vote
1answer
47 views

Uniqueness proof of binary representation

I'm having trouble understanding this proof of uniqueness of binary representation of integers: Suppose there exists an integer $n$ with two different binary representations. Let these be: ...
10
votes
2answers
118 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
3
votes
1answer
70 views

Schnirelmann Density: if $d(A) + d(B) \ge 1$, does it follow that $d(A+B)=1$

I am still trying to get my head around the basic properties of Schnirelmann Density. If I'm reading PlanetMath.org correctly, it states that if $d(A) + d(B) \ge 1$, then $d(A+B)=1$ Here's the exact ...
4
votes
1answer
66 views

Summation of a floored square root

I am working on a little something and have hit a roadblock of sorts. I have arrived at this equation:$$\sum_{n=1}^{r}\left\lfloor\sqrt{2nr-{n}^{2}}\right\rfloor$$ I am attempting to find some way of ...
2
votes
0answers
76 views

Need to find a better algorithm to solve a project euler problem dealing with coprime pairs.

I've been working on this for a while and found several solutions so far, but none are fast enough to find the necessary $S(10^7)$. Here is the question: For an integer $M$, we define $R(M)$ as ...
6
votes
2answers
109 views

$a_1=1,a_{n+1}=\frac{n}{a_n}+\frac{a_n}{n}$. Prove that for $n\ge4$, $\lfloor{a_n^2}\rfloor=n$

Define a sequence $\left\lbrace a_{n}\right\rbrace$ by $\displaystyle{a_{1} = 1\,,\ a_{n + 1} = {n \over a_n} + {a_n \over n}.\quad}$ Prove that for $n \geq 4,\,\,\left\lfloor ...
0
votes
0answers
51 views

sum of digits puzzle

A center-coordinator was supervising the arrangements being made for a public exam. Tables , meant for individual candidates, were arranged in a number of columns with each column having the same ...
0
votes
0answers
139 views

Is there a elementary way to prove $\zeta(2)=\frac{\pi^2}{6}$

The proof in the Wikipedia is still much complicated, can any one provide a really simple way to prove this.
1
vote
2answers
851 views

the nth root of n!?

I am playing around with the root/ratio test to practice with series. I just showed that $\sum \frac{1}{n!}$ converges by using the ratio test. I decided to see how things would go with the root test ...
2
votes
1answer
245 views

Need help with a zeta-like function?

Some time ago I found interesting modifications to Euler's prime product that produces a square number and its square root. The parts that were still unknown were the corresponding sums. I have ...
1
vote
1answer
53 views

On the rational Beatty sequence

Let $S(p/q, b) = \{[pn/q + b]|n\in\mathbb{Z}\}$, where $p, q$ are coprime positive integers and $b$ is any integer, be a rational Beatty sequence. I can't see why the following conclusion is true: ...
3
votes
2answers
140 views

On proving the convergence of $1/n^2\sum_{1\le k\le n}\varphi(k)$

Let $$\Phi_n=\frac{1}{n^2}\sum_{k=1}^n\varphi(k).$$ How one can show that $\Phi_n$ is convergent sequence? (Here, $\varphi$ denotes the Euler's totient function.) And please, without any monster ...
1
vote
2answers
117 views

Divisibility of sequence

Let the sequence $x_n$ be defined by $x_1=1,\,x_{n+1}=x_n+x_{[(n+1)/2]},$ where $[x]$ is the integer part of a real number $x$. This is A033485. How to prove or disprove that 4 is not a divisor of any ...
7
votes
1answer
183 views

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Transcendental?

Is $\sum_{k=0}^{\infty}\frac1{2^{k^2}}$ rational? Clearly this series is convergent (compare to geometric series with ratio 1/2). I'm sure it's irrational since a rational number written in base 2 ...
5
votes
2answers
209 views

Is Collatz' conjecture the only stable solution of its type?

The Collatz Conjecture is well known with the sequence $$f(n) = \begin{cases} n/2 &;\text{if } n \equiv 0 \pmod{2}\\ k\,n+1 &; \text{if } n\equiv 1 \pmod{2} \end{cases}$$ and $k=3$; the ...
1
vote
2answers
72 views

Irrationality proof by rational approximations

Assume we have a sequence of rational numbers $\left(\frac{p_n}{q_n}\right),$ where $\gcd(p_n,q_n)=1, \ \forall n \in \mathbb N$. We know that $$\lim_{n\to\infty} \left(\frac{p_n}{q_n}\right)= x$$ ...
2
votes
2answers
110 views

Is Fibonacci sequence the minimum of unique pairwise sum sequence?

Let $(a_n)_{n=1}^\infty$ be a strictly increasing (condition added per earlier answer of Amitesh Datta) sequence of natural numbers where all pairwise element sums are unique. Can anyone prove or ...
1
vote
2answers
199 views

“Interesting” Sequences

Well, here's a question i myself made up and i thought it's interesting if i share it with everyone. We call a sequence of natural numbers (for example $a$) Interesting if (all three must be true): ...
4
votes
2answers
220 views

summation of consecutive natural numbers does not end in 7,4,2,9

I calculated sum of n consecutive natural numbers where n = 1 to 100 .What I mean is $$\sum_{n=1}^{1}n = 1 $$ $$\sum_{n=1}^{2}n = 3 $$ $$\sum_{n=1}^{3}n = 6 $$ And I got answers and noticed that ...
27
votes
2answers
953 views

Limit of recursive sequence $a_{n+1} = \frac{a_n}{1- \{a_n\}}$

Consider the following sequence: let $a_0>0$ be rational. Define $$a_{n+1}= \frac{a_n}{1-\{a_n\}},$$ where $\{a_n\}$ is the fractional part of $a_n$ (i.e. $\{a_n\} = a_n - \lfloor a_n\rfloor$). ...
0
votes
1answer
122 views

Pascal's other triangle

Just a brainteaser question: Can you identify the generator of the following pattern of numbers?      Remark on any interesting patterns you see in the triangle.
8
votes
7answers
384 views

Why is the Fibonacci ratio though a decreasing function, it is alternating and decreasing?

I tried to find the ratio of consecutive terms of the Fibonacci series and found that it is a decreasing function and it converges . I tried it though a small code piece in python so that I can have a ...
3
votes
2answers
326 views

Linked Arithmetic progression and Harmonic progression

I would like to give some introduction about the origin of my doubt and then put forth my doubt , so that people who attempt answering will know the context . ...