# Questions tagged [digits]

The digits tag has no usage guidance.

35
questions

13
votes

1
answer

521
views

### Can we compute the first $n$ digits of $\pi$ in $F(n)$ time?

I've seen various fast algorithms for computing the first few, or directly the $n$-th, digits of $\pi$.
However, it seems to me that all these algorithms assume (see last sentence here) that there are ...

15
votes

2
answers

1k
views

### The parity of the maximal number of consecutive 1s in the binary expansion of an integer

For an integer $n$, let $\ell(n)$ denote the maximal number of consecutive $1$s in the binary expansion of $n$. For instance,
$$ \ell(71_{10}) = \ell(1000111_2) = 3. $$
Consider the set $E$ of all ...

24
votes

0
answers

905
views

### 0's in 815915283247897734345611269596115894272000000000

Is 40 the largest number for which all the 0 digits in the decimal form of $n!$ come at the end?
Motivation: My son considered learning all digits of 40! for my birthday. I told him that the best way ...

3
votes

2
answers

686
views

### Density of the set of numbers whose sum of digits is prime

Let $A$ be the set of numbers whose sum of digits is prime (http://oeis.org/A028834).
I would like to know if $A$ has zero natural density, that is, if $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...

6
votes

1
answer

767
views

### How to explain this prime gap bias around last digits?

My question is related to this article by Oliver and Soundararajan (article about a bias in the distribution of the last digits of consecutive prime numbers).
After trying some python experimental ...

1
vote

1
answer

317
views

### Problem related to inequality of sum of digits of power sum

Let $D$ be the function define as $D(b,n)$ be the sum of the base-$b$ digits of $n$.
Example: $D(2,7)=3$ means $7=(111)_2\implies D(2,7)=1+1+1=3$
Define $S(a,m)=1^m+2^m+3^m+...+a^m$ where $a,m\in\...

2
votes

1
answer

81
views

### Partitioning integers into two parts and exploring relationships with positional numeral systems

I asked this question in Mathematics StackExchange (link) about a month ago, but I have received no answer. It is about the following problem:
Problem:
Are there sets $A,B$ of integers such that $A\...

0
votes

1
answer

111
views

### The series $\sum_{n=1}^\infty {2n\brace n}^{-{2n\brace n}}$ and $\sum_{n=1}^\infty (2n)_{n}^{-(2n)_{n}}$ in the context of normal numbers

In this ocassion we consider the followgin series that involve ${n\brace k}$ the Stirling number of the second kind and $(n)_k$ the Pochhammer symbols. I've known from an informative point of view ...

2
votes

1
answer

109
views

### Measure of real numbers with converging average over binary digits

Consider the unit interval $[0,1]$, and by digits of $x\in[0,1]$ I mean its binary digits after the separator with no 1-period.
If $x_1,x_2,x_3,...$ are the digits of $x$, then consider the $k$-th ...

1
vote

0
answers

562
views

### Are either $\pi + e$ or $\pi e$ transcendental if we add or multiply digit-wise? [closed]

Since $x^2 - (e + \pi)x + e \pi = (x - \pi)(x - e)$ has transcendental roots, we know that the coefficients are not both rational, and not even algebraic (see comment by José).
My question is, can we ...

1
vote

1
answer

189
views

### Runs of consecutive numbers that are not relatively prime to their digital sum

It is well known that there can be at most 20 consecutive integers (in base 10) that are divisible by their digital sum, so called Harshad or Niven numbers.
How long can a run of consecutive ...

0
votes

0
answers

77
views

### Generating the digits in a base system by repeated multiplication of a number

The first 15 terms of the sequence {a_i} = 2^i are 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8192, 16384, 32768. All of the digits in base-10, i.e. {0, ...

2
votes

1
answer

200
views

### The number of numbers no greater than n that are divisible by all their suffixes

My question: what a formula for finding the number of numbers no greater than n that are divisible by all their suffixes.
e.g: 5, 25, 125, 0125, 70125 are divisors of 70125.
refinement: $\overline{0....

0
votes

1
answer

226
views

### Does a sequence of primes defined like this exists?

Does there exist a strictly increasing sequence of primes $(q_i)_{i \in \mathbb N}$ such that $\text {ds} (\prod_{k=1}^l q_k)$ is prime for every $l \in \mathbb N$?
Here $\text{ds}(n)$ denotes a ...

3
votes

0
answers

158
views

### A recursion for the total number of 1's in binary expansions of the first natural numbers?

Let $$a(n)=a(2^k-n)+k(n-2^{k-1})$$ for $$1 \leqslant {2^{k - 1}} < n \leqslant {2^k}$$
with initial values $a(0)=0, a(1)=0, a(2)=1.$
The first values are $0,0,1,2,4,5,7,9,12,13,15,\dots.$
...

2
votes

0
answers

74
views

### Powers of special class of positive integers whose representation in a base consists of digits only powers of that integer

For an integer $m \in (\sqrt {10} , 10)$ , define $A_{10,m}:=\{n \in \mathbb N : m^n=\sum_{j=0}^k 10^j m^{n_j} ; n_j=0 $ or $1; k \ge 0\}$ . So , $A_{10,m}$ is the set of those natural numbers , ...

34
votes

2
answers

2k
views

### Is the sum of digits of $3^{1000}$ divisible by $7$?

Is the sum of digits of $3^{1000}$ a multiple of $7$?
The sum of the digits of $3^{1000}$ can be computed using a computer. It is equal to $2142$, so the answer is positive.
Is there a short proof ...

6
votes

2
answers

616
views

### Does this sequence of ratios of digit sums have a limit?

I asked this question a few hours ago on MathStackExchange and there it received some attention but we still do not have a proof so I decided to ask it here also, in an unchanged form, and here it is:
...

5
votes

2
answers

910
views

### Is there a Bailey–Borwein–Plouffe (BBP) formula for e? [duplicate]

I recently used Bailey–Borwein–Plouffe formula to implement a π digit generator. Now I also want to implement an e digit generator, for the Euler number.
I've ...

6
votes

0
answers

187
views

### Choice of digits for extensions of $\mathbb{Q}_p$

I am interested in writing (in base $p$) elements of the maximal unramified extension $\mathbb{Q}_p^{\mathrm{unr}}$ of $\mathbb{Q}_p$, or (its completion) the field $\mathrm{W}(\mathbb{F}_p^{\mathrm{...

4
votes

1
answer

286
views

### Can we always attain another prime via inserting digits between the digits of a fixed prime?

The sequence OEIS A080437 is
For n > 10, let m = n-th prime. If m is a k-digit prime then a(n) = smallest prime obtained by inserting digits between every pair of digits of m.
I don't see why this ...

1
vote

0
answers

70
views

### On the sum of digits of primes in binary form [duplicate]

Let $s_2(m)$ be the sum of digits of $m$ in binary form.
I would like to ask the following question:
Is it true that for every $n\in \mathbb{N}$ there is at least one
prime $p$ which has $s_2(...

1
vote

0
answers

720
views

### Kaprekar's mapping fixed points

Jens Kruse Andersen in his comment in OEIS's A099009 noticed 3 families of numbers among Kaprekar's fixed mapping points (otherwise known as kernels of the Kaprekar's routine):
"Let $d(n)$ denote $n$...

8
votes

0
answers

399
views

### Zero's in the decimal representation of powers of 3

This looked like an easy exercise, when a friend of mine asked me if I know a way to prove that the decimal representation of $3^k$ always contains a zero for $k\ge k_0$, but the more I think about ...

6
votes

3
answers

663
views

### sum of binary and ternary digits

A problem in group theory (indices of imprimitive groups) gives rise to the following conjectures in number theory. Suppose a positive integer $n$ has binary and ternary expansions $n=\sum_{k\...

6
votes

1
answer

645
views

### Optimal lower bounds for the sum of digits in base $b$

Let $b \geq 2$ be an integer and let $s_b(n)$ be the sum of the digits of the base-$b$ representation of the nonnegative integer $n$
(e.g., $s_{10}(726)=7+2+6$). From the weak law of large numbers, it ...

12
votes

0
answers

527
views

### Power series defined by Witt vectors / Teichmüller representatives of p-adics

Let $K$ be $\mathbb{Q}_p$ for some prime $p$ (or more generally an unramified extension $W(\mathbb{F}_q)$ of $\mathbb{Q}_p$). If $\xi \in K$, we can write it in a unique way in the form $\sum a_i p^i$...

3
votes

1
answer

192
views

### Normality property of powers of integers?

Inspired by this question, is there some conjecture stating that
$$
\limsup_{n \to \infty} \frac{d_j(2^n)}{dc(2^n)} = \frac{1}{10}
$$
where $d_j(m)$ counts the number of $j$s in the digits of $m$,
and ...

4
votes

2
answers

3k
views

### Number of 1 in binary representation of n

Let $1(n)$ be the number of digits $1$ in binary representation of number $n$.
For example, $13=1101_2$ so $1(13)=3\\$
Is there explicit form of $\,\,\sum{1(i)x^i} $?
I checked OEIS and didn't find ...

15
votes

0
answers

2k
views

### Distribution of digits of $pq$-adic idempotents (aka "automorphic numbers")

Let $p$ and $q$ be distinct primes. By the ring of $pq$-adic integers I mean the ring $\mathbb{Z}_{pq} := \varprojlim \mathbb{Z}/(pq)^n\mathbb{Z}$ which is obviously isomorphic to $\mathbb{Z}_p \...

1
vote

2
answers

307
views

### Square and reversed integer

For all $n=\overline{a_k a_{k-1}\ldots a_1 a_0} := \sum_{i=0}^k a_i 10^i\in \mathbb{N}$, where $a_i \in \{0,...,9\}$ and $a_k \neq 0$,
we define $f(n)=\overline{a_0 a_1 \ldots a_{k-1} a_k}= \sum_{i=0}...

3
votes

0
answers

1k
views

### sum of digits in different bases

Given a natural number, What is the maximal natural number below it, whose sums of digits in base 10 and base 2 are the same? Is there a clever algorithm to do this aside from the brute force search? ...

2
votes

1
answer

695
views

### Here is a generalization of n-ary base notation for numbers. Surely unoriginal. Anybody know where to find literature on it?

If $f:\mathbb{N}\to\mathbb{N}$ is any strictly increasing function with $f(0)=1$, define the base $f$ notation for natural numbers inductively as follows:
$0$ is represented as $()$ (the empty ...

9
votes

1
answer

716
views

### Lower bound on # of nonzero digits in ternary expansions of powers of 2?

Does anyone know of any lower bounds on the number of nonzero digits that appear in powers of 2 when written to base 3? (Other than the easy "If it's more than 8 it has to have at least 3.") I know ...

47
votes

5
answers

5k
views

### Can $N^2$ have only digits 0 and 1, other than $N=10^k$?

Pablo Solis asked this at a recent 20 questions seminar at Berkeley. Is there a positive integer $N$, not of the form $10^k$, such that the digits of $N^2$ are all 0's and 1's?
It seems very unlikely,...