All Questions
Tagged with sequences-and-series decimal-expansion
72
questions
9
votes
1
answer
303
views
Is there a non-trivial arithmetic progression of positive integers such that every number contains the digit $2?$
Let $X:=\{$ positive integers that contain the digit $2\}$
For fixed $m,n\in\mathbb{N},$ define the A.P. $S_{m,n:=}\ \{m,\ m+n,\ m+2n, \ldots\}\ .$
I am interested in $S_{m,n}\cap X,$ and $S_{m,n}\cap ...
- 20.7k
1
vote
0
answers
30
views
Existence of $n$ where $S_b(n^k) \equiv r \pmod{M}$ where $S_b$ denotes sum of digits in base $b$
Let $b, k, M \in \mathbb{N} \setminus \{1\}$, $r \in \{0, 1, \dots M-1\}$ and $S_b: \mathbb{N}_0 \rightarrow \mathbb{N}_0 $ denote the method which outputs the sum of digits of its input in base $b$.
...
- 1,191
2
votes
2
answers
128
views
Does $\sum\limits_{n=1}^\infty\frac{1}{\text{Sum of permutations of digits of }n}$ converge?
Hopefully the following chart explains some things:
$$\begin{array}{|l|l|} \hline
n & \frac{1}{\text{Sum of permutations of digits of }n} \\ \hline
1 & \frac{1}{1} \\ \hline
2 & \frac{1}{2}...
- 1,724
5
votes
3
answers
352
views
Arbitrary decimal value of $A(n)=\left(\frac{11}{10}\right)^n$
For $n\in\mathbb{Z}$ consider the number
$$A(n)=\left(1+x\right)^n\bigg{|}_{x=\frac{1}{10}}=\sum_{k=0}^\infty\binom{n}{k}10^{-k}$$
which we have expanded by the Taylor series. It is found that
$$a_1=\...
- 51
0
votes
0
answers
56
views
If $\{n_k\}$ is the set of natural numbers with no 0 in their decimal expansion, $\sum_{k=1}^\infty \frac{1}{n_k}$ converges to a number less than 90 [duplicate]
Let ${\{n_1,n_2,…\} }$
be the set of natural numbers that do not use the digit 0
in their decimal expansion. Then, the series
$\sum_{k=1}^\infty \frac{1}{n_k}$
converges to a number less than 90.
Is ...
- 389
0
votes
2
answers
841
views
Is there any perfect power in the sequence $12,123,1234,12345,...$?
Inspired from the question Is there any perfect square in the sequence $12,123,1234,12345,...$?, there is no perfect square other than $1$ in the sequence of Smarandache numbers. But I wonder if are ...
- 628
4
votes
2
answers
203
views
Proof of $\inf\left \{ \frac{\mathrm{d} (n^2)}{\mathrm{d} (n)} \; \bigg| \; n \in \mathbb{N} \right \}=0$, where $d(n)$ is the sum of digits of $n$
So I wanted to find the infimum of the set described in the title, and I'm pretty confident on what the subsequence should be to ensure a 0 infimum.
But a proof of this eludes me. I tried some funky ...
- 506
3
votes
2
answers
94
views
The sum of the numbers from $100$ to $999$ that do not have the digit $0$ as well as do not have repeated digit.
Considering the numbers from $100$ to $999$. Excluding numbers that have the digit $0$, also excluding numbers that have repeated digit. What is the sum of remaining numbers?
That is, we need to find
$...
- 5,450
0
votes
1
answer
102
views
In any base $b$, $\forall n\in\mathbb{N}$ built from digits $\{1,2\}$, if you tally distinct digits in those $n^2$, finitely many have $\leq b/2$
E.g. let $f(n)$ be the $n$th largest natural number consisting of only digits $\{1,2\}$, written in base-10.
It appears from early data that for $1\leq k \leq 9$, there is a maximum $n$ such that the ...
- 6,014
5
votes
2
answers
165
views
Can we define addition of numbers which are **NOT** eventually all zero as we go to the left?
I am struggling to define addition of objects which are similar to decimal-expansions.
In this post, we refer to the decimal-expansion-like things as "wumbers".
Our goal is to write ...
- 1,104
2
votes
1
answer
130
views
Eventually-prime decimal expansions
Let $w$ be a right-infinite word over the alphabet $A = \{ 0, 1, \dots, 9\}$, with a distinguished decimal point after at most finitely many symbols from the left (i.e. $w$ is in $A^\ast . A^\omega$). ...
- 193
1
vote
2
answers
450
views
Is $\pi$ in the infinite set $ \{ 3, 3.1, 3.14, 3.141, 3.1415, ...\} $?
Does $\pi$ exist in the following infinite set. Apologies for lack of set notation, and i'm hoping its not necessary to help me understand the nature of infinite decimal expansion.
I know sets are ...
4
votes
3
answers
156
views
What is the sum of natural numbers that have $5$ digits or less, and all of the digits are distinct?
$1+2+3+\dots+7+8+9+10+12+13+\dots+96+97+98+102+103+104+\dots+985+986+987+1023+1024+1025+\dots+9874+9875+9876+10234+10235+10236+\dots+98763+98764+98765=$
The only thing I can do is to evaluate a (bad) ...
- 5,450
7
votes
3
answers
140
views
When is $X/R(X)$ an integer where $R(X)$ is the reverse of an integer $X$?
My question concerns reverse numbers (e.g. $1234 → 4321$).
Is it possible to find integer solutions greater than $1$ for such numbers when you take their ratio? I am not interested in trivial ...
- 71
3
votes
1
answer
971
views
Prove that every real number $x \in [0,1]$ has a "certain" binary representation
I want to prove that every real number $x \in [0,1]$ has binary representation in the following way:
Let $B$ denote the set of all sequences $b:\mathbb{N} \rightarrow\{0,1\}$. Consider then $f:B\...
- 297