# Questions tagged [digits]

The tag has no usage guidance.

33 questions
Filter by
Sorted by
Tagged with
814 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 ...
587 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 $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...
647 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 ...
276 views

101 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 ...
104 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 ...
550 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 ...
167 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 ...
68 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, ...
189 views

278 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 ...
66 views

544 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 ...
472 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 $\sum a_i p^i$...
185 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 ...
2k 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 ...
2k views

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? ...
689 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 ...
### 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,...