2
votes
1answer
47 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
37 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 $c_0 \neq 2$?
-3
votes
0answers
55 views

Which of the following is correct?

Let $$X = \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$$ Then find the correct option. (A) $X < 1$ (B) $X > \frac{3}{2}$ (C) $1 < X < \frac{3}{2}$ (D) ...
0
votes
2answers
55 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
25 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
107 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
56 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
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 ...
9
votes
2answers
133 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
108 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
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 ...
3
votes
1answer
170 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
39 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
107 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
63 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
59 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
70 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 ...
5
votes
2answers
104 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
45 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
125 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.
2
votes
1answer
243 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 ...
0
votes
1answer
44 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
136 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
114 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
178 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
200 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
65 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
107 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
185 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
163 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
911 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
108 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
350 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
308 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 . ...
1
vote
3answers
87 views

Sequences of integers with lower density 0 and upper density 1.

It is possible to construct a sequence of integers with lower density 0 and upper density 1? where lower and upper density means asymptotic lower and upper density (cf. References on density of ...
3
votes
1answer
152 views

The sum of the squares of the prime factors

Define the number $d_n$ as $$d_n = p_1^2 + p_2^2 + \dots + p_k^2$$ where $p_1, p_2, \dots, p_k$ are the $k$ prime factors of $n$. For example, $d_{220} = 2^2 + 2^2 + 5^2 + 11^2 = 154$ since $220 = 2 ...
2
votes
3answers
44 views

Help with a step in rearranging this problem

i'm working through the proof of this theorem; If $x$ is any real number other than $1$, then $$\sum_{j = 0}^{n -1} x^j = 1 + x + x^2 + \cdots + x^{n-1} = \frac{x^n-1}{x-1}$$ But i'm struggling with ...
0
votes
2answers
48 views

Translation of; If X is any real number other than 1, then…

i've just started reading a book on number theory and am trying to follow along with the example proofs of theorems. I've not had too much trouble once I have managed to "translate" the mathematical ...
0
votes
1answer
80 views

When is an infinite sequence of integers purely deterministic with no randomness involved?

I see in literature very different descriptions of what is a deterministic system such as: "... a system in which no randomness is involved in the development of future states of the system...>>>" I ...
5
votes
4answers
411 views

Find a function that gives this sequence: $+1,+1,-1,+1,+1,-1,-1,+1,+1,+1,-1,-1,+1,-1,-1,…$

I start with a string $S_1=1$ then the $(n+1)$-th string is $S_{n+1}=\{ S_n,+1 ,-(S_n)\}$ if $S_j=\{s_1,s_2,s_3,..., s_i\}$ then $-(S_j)$ is defined as $-(S_j)=\{-(s_i), -(s_{i-1}),..., -(s_3), ...
1
vote
1answer
49 views

REVISTED$^1$ - Order: Modular Arithmetic

Relevant Literature: Question: Observe that $2^{10}=1024≡−1 \pmod{25}$.Find the order of $2$ modulo $25$. Thoughts: Direct answers are OK, but I'd like to know if I'm right that what I'm really ...
4
votes
1answer
283 views

Generalizing Ramanujan's sum of cubes identity?

Ramanujan's sum of cubes identity is defined by the generating functions, $$\begin{aligned} \sum_{n=0}^\infty a_n x^n &= \frac{1+53x+9x^2}{R_1}\\ \sum_{n=0}^\infty b_n x^n &= ...
0
votes
1answer
62 views

How many of the prime factors of an even number are 2?

I'm trying to concisely write an expression for a sequence which flips a switch $n$ times where $n$ is the number of times 2 divides evenly into $n$, so the first 24 values are: ...
1
vote
3answers
88 views

Fibonacci series in 0-Even-Odd-Even-Odd-N series up to N

Another question from the test for the Normale of Pisa: Consider the series $S_n$ of integer numbers repeteandly even - odd - even - odd that start with 0 and finish with n, so with n = 3 we get 2 ...
6
votes
1answer
85 views

How to prove $\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$ where $a_i\in\mathbb N$ and $a_i\lt a_{i+1}$?

Let $a_1,a_2,\ldots ,a_n\in\mathbb N$ and $a_1\lt a_2\lt\cdots\lt a_n$. Then how to prove $$\sum_{i=1}^{n-1}\frac{1}{\operatorname{lcm}(a_i,a_{i+1})}\lt1$$ Thanks in advance
0
votes
0answers
45 views

Minimum of a linear congruence sub-sequence

I have the following little problem : let $a,b,u,v$ be four given integers with $\gcd(a,b)=1$. Now I would like to find the minimum of the linear congruence subsequence $\{ax \pmod b : u \le x \le ...
38
votes
7answers
856 views

Problems regarding $\{x_n \}$ defined by $x_1=1$; $x_n$ is the smallest distinct natural number such that $x_1+…+x_n$ is divisible by $n$.

Let me denote a sequence of distinct natural numbers by $x_n$ whose terms are determined as follows: $x_1$ is $1$ and $x_2$ is the smallest distinct natural number $n$ such that $x_1+x_2$ is divisible ...
6
votes
2answers
900 views

Multiples of an irrational number forming a dense subset

Say you picked your favorite irrational number $q$ and looking at $S = \{nq: n\in \mathbb{Z} \}$ in $\mathbb{R}$, you chopped off everything but the decimal of $nq$, leaving you with a number in ...
0
votes
1answer
39 views

Progressions with variable density that can be described in constant space?

Say we have an arithmetic progression in $Z_n$ like $3, 6, 9, 12, ...$ etc. If you move a sliding window of at least 3 values over the progression the 'density' in that subset compared to if the ...