Tagged Questions

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Powerful numbers in Pell solutions (or, more generally, any Lucas sequence)

There are several definitive results regarding perfect powers in the Pell numbers ā e.g., the only perfect power is $P_7=169=13^2$. On the other hand, when it comes to powerful numbers, I've only ...
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What notation to use for a sequence of integers that end with digit 5?

I need to solve a low high school home work and I ask a question about the most correct notation. The problem is to build a set of circles with $r$ and $d$ such that $d=5, 15, 25, 35,...d_{+_1}$ and ...
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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'' ...
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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.
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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 ...
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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 $A = \{2nā1\ |\ n\in\mathbb N\}$. Let $B=\mathbb Z_{\ge 0}-A$. Let $C$ be the set that ...
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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: $$x_1x_2x_3=āy_1y_2y_3 \quad \text{and} \quad x^2_1+x^2_2+x^2_3=y^2_1+y^2_2+y^2_3$$ nd that ...
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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 ...
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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 ...
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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.
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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 $$\sum\limits_{n=1}^{\infty}\frac{a(n)}{n(n+1)}.$$
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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 ...
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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 $9$ ...
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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.
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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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $$\lim_{n\to\infty} \left(\frac{p_n}{q_n}\right)= x$$ ...
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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 ...
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): ...
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 ...