Questions tagged [digits]

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How to find A(i, d)?

Let $s(n)$ denote the digit sum of a natural number $n$. For $i, d\in \mathbb{N}$ define $$A(i, d) = \limsup_{m\to \infty}\frac{|\{n\leq m | s(n)\equiv i\mod d\}|}{m}.$$ Compute $A(i, d)$ for all $i, ...
Drrd's user avatar
  • 11
4 votes
0 answers
434 views

There are infinitely many prime which have arbitrary large gap in their digits in particular base expansion

Consider $m$ and $r$ is any fixed positive integer and $t$ is a variable $(t=0,1,2,3,...)$. Below, $[a]$ denotes the greatest integer function of $a$ (or floor function). Claim 1 : There exists a ...
Pruthviraj's user avatar
6 votes
1 answer
263 views

Sequence of digits of powers of two

Elementary number theory tells us a lot about the final digits of the powers of two, and ergodic theory (more specifically the theory of equidistribution of points in the orbit of an irrational ...
James Propp's user avatar
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2 votes
1 answer
64 views

Maximum product of digits of a perfect power

Today I found an interesting problem on the Internet - to find the exact power that has the largest possible product of digits. Of course, we know that there are arbitrarily large squares that do not ...
Vanya Borisyuk's user avatar
-4 votes
1 answer
340 views

Multiplicative Persistence - Highest persistence found? [closed]

tried to ask on the math reddit but got deleted due to my account being new. Is the record for highest multiplicative persistence found still 11? As I may have just found a number with persistence of ...
mwt2212's user avatar
  • 21
2 votes
1 answer
94 views

Consecutive prime numbers in permutations of digits of the first consecutive positive integers

I have been toying for a while with the study of: in how many distinct primes and of which size can we divide permutations of digits of the first positive integers? In this post I studied how many ...
Juan Moreno's user avatar
3 votes
1 answer
133 views

Golden ratio base

Let $\phi$ be the golden ratio and look at real numbers as expansions in digits from base $\phi + 1$. Has this base been considered or studied anywhere? Note that integers in this base are palindromes ...
Maarten Havinga's user avatar
0 votes
0 answers
70 views

Lowest asymptotic bound to $4^n - 2v_n^2$ where $v$ is an odd integer, $n$ fixed

The general problem is this. I try to find a positive integer $\delta_n$ such $qv_n^2 +\delta_n = p\cdot 4^n$. More precisely, I am looking for a lower bound (depending on $n$) for $\delta_n$ as $n\...
Vincent Granville's user avatar
13 votes
1 answer
555 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 ...
domotorp's user avatar
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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 ...
Jakub Konieczny's user avatar
24 votes
0 answers
981 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 ...
domotorp's user avatar
  • 18.2k
3 votes
2 answers
779 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 $$\lim_{n \to +\infty} \frac{A(n)}{n} = 0,$$ ...
dache1771's user avatar
6 votes
1 answer
845 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 ...
Thierry Boulord's user avatar
1 vote
1 answer
358 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\...
Pruthviraj's user avatar
2 votes
1 answer
87 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\...
C. Fujinomiya's user avatar
0 votes
1 answer
127 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 ...
user142929's user avatar
2 votes
1 answer
113 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 ...
M. Winter's user avatar
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1 vote
0 answers
568 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 ...
user avatar
1 vote
1 answer
205 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 ...
Bernardo Recamán Santos's user avatar
0 votes
0 answers
82 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, ...
Matthew Lim's user avatar
2 votes
1 answer
227 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....
Martin Leshko's user avatar
0 votes
1 answer
247 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 ...
Shalom's user avatar
  • 513
3 votes
0 answers
165 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 $0,0,1,2,4,5,7,9,12,13,15,\dots.$ ...
Johann Cigler's user avatar
2 votes
0 answers
75 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 , ...
user avatar
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 ...
Lezraf's user avatar
  • 443
6 votes
2 answers
852 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: ...
user avatar
5 votes
2 answers
999 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 ...
user avatar
6 votes
0 answers
208 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{...
Gro-Tsen's user avatar
  • 29.6k
4 votes
1 answer
297 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 ...
joro's user avatar
  • 24.2k
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(...
Konstantinos Gaitanas's user avatar
1 vote
0 answers
737 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 $n$...
Alex's user avatar
  • 335
8 votes
0 answers
414 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 ...
Omran Kouba's user avatar
6 votes
3 answers
687 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\...
Glasby's user avatar
  • 1,961
6 votes
1 answer
693 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 ...
user avatar
12 votes
0 answers
571 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 $\sum a_i p^i$...
Gro-Tsen's user avatar
  • 29.6k
3 votes
1 answer
196 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 ...
Per Alexandersson's user avatar
5 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 ...
Radmir's user avatar
  • 443
17 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 \...
Gro-Tsen's user avatar
  • 29.6k
1 vote
2 answers
323 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}...
user12806's user avatar
  • 663
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? ...
Hans's user avatar
  • 2,169
2 votes
1 answer
705 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 ...
Sam Alexander's user avatar
9 votes
1 answer
732 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 ...
Harry Altman's user avatar
  • 2,535
51 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,...
Scott Morrison's user avatar