Questions tagged [digits]
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33
questions
24
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0answers
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 ...
3
votes
2answers
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 $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...
6
votes
1answer
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 ...
1
vote
1answer
276 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
1answer
79 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
1answer
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 ...
2
votes
1answer
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 ...
1
vote
0answers
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 ...
1
vote
1answer
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 ...
0
votes
0answers
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, ...
2
votes
1answer
189 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
1answer
221 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
0answers
154 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
0answers
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 , ...
35
votes
2answers
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
2answers
401 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
2answers
811 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
0answers
165 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
1answer
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 ...
1
vote
0answers
66 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
0answers
713 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
0answers
382 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
3answers
622 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\...
5
votes
1answer
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 ...
11
votes
0answers
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 $\sum a_i p^i$...
3
votes
1answer
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 ...
4
votes
2answers
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 ...
13
votes
0answers
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
2answers
283 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
0answers
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
1answer
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 ...
9
votes
1answer
698 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 ...
45
votes
5answers
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,...
