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Questions tagged [digits]

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2
votes
1answer
128 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
217 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
145 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
72 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
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
396 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
532 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
143 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
265 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
53 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
686 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
350 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
547 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
274 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
351 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
177 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 ...
11
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
250 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
883 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
680 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 ...
8
votes
1answer
666 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 ...
44
votes
5answers
4k 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,...