25
votes
Accepted
Total digit sum in distinct bases grow unboundedlly
As observed in comments, it suffices to show that for fixed $m,k$, there are only finitely many solutions in non-negative integers to the equation
$$ a^{x_1} + \dots + a^{x_m} = b^{y_1} + \dots + b^{...
- 114k
25
votes
Accepted
The parity of the maximal number of consecutive 1s in the binary expansion of an integer
Perhaps surprisingly, the random variable $\ell(n)$ (with $n$ drawn uniformly from $[0,N)$) concentrates too much around $\log_2\log_2 N$ (where $\log_2$ denotes the logarithm to base $2$) to have a ...
- 114k
18
votes
Accepted
Can we compute the first $n$ digits of $\pi$ in $F(n)$ time?
Mahler proved(1) that for any $p,q$, $\left | \pi -\frac{p}{q} \right| > \frac{1}{q^{42}}$. It follows that if one can compute $2^{ 42n} \pi$ to within an error of at most $1$, one can compute the ...
- 148k
16
votes
Accepted
Density of the set of numbers whose sum of digits is prime
Yes, $A(n)$ has zero natural density. It suffices to prove this for $n$ which is a power of $10$.
and it is possible to make this more precise. To see this, first let $n=10^k$ and note that for $X$ ...
- 14.2k
12
votes
Multiplicative Persistence - Highest persistence found?
First, to your direct question: "Is the record still 11", the answer is "yes", according to a variety of sources (MathWorld, Wikipedia, OEIS A003001). If anyone has found a new ...
- 4,219
11
votes
Is the sum of digits of $3^{1000}$ divisible by $7$?
Middle digits of the numbers $3^n$ are unpredictable. At least it is too hard for current techniques to say anything about them. It means that the their sum is unpredictable as well. Some good random ...
- 12.3k
10
votes
Accepted
How to explain this prime gap bias around last digits?
Yes, it can be analyzed in the same way. Since the effects you're measuring are similar to each other, I'm only going to address the first one, cumulative prime gaps after primes with a fixed last ...
- 116
8
votes
Is the sum of digits of $3^{1000}$ divisible by $7$?
Not an answer, but a series of considerations.
One expects not only the digit sum of 3^n to be a multiple of nine (for integral n greater than 1) but also for the string of digits (in the decimal ...
- 13k
8
votes
Accepted
Measure of real numbers with converging average over binary digits
The set $\{x\in[0,1]:\lim_k s_k(x)=\frac12\}$ is the complement of a null set. This is an instance of the strong law of large numbers.
- 73.2k
8
votes
The parity of the maximal number of consecutive 1s in the binary expansion of an integer
Here is a small visualisation of the values of this function.
I have taken $n \in \{1,2,\ldots,512\}$ and calculated the proportion of the numbers $0 \leqslant x < 2^n$ that have a maximum 1-run of ...
- 12.7k
6
votes
Density of the set of numbers whose sum of digits is prime
The following paper by Glyn Harman
(Counting primes whose sum of digits is prime.
J. Integer Seq. 15 (2012), no. 2, Article 12.2.2, 7 pp)
studies a more complicated situation: count primes $p \leq X$,
...
- 2,705
6
votes
Is there a Bailey–Borwein–Plouffe (BBP) formula for e?
I wish I had 50+ reputation so I could comment : I visited your website and I saw that you are not actually using what the BBP is famous for namely extracting the $n$-th digit in base-16. Instead you ...
- 221
4
votes
Accepted
Runs of consecutive numbers that are not relatively prime to their digital sum
As long as you wish. Let $s(n)$ denote the sum of decimal digits of $n$.
Lemma. For any positive integer $k$ there exist distinct prime numbers
$p_1<p_2<\ldots<p_k$ such that $p_1>5$ and $...
- 109k
4
votes
Does this sequence of ratios of digit sums have a limit?
If the limit exists, it of course equals 1. Indeed, $A(m):=a(m)a(m+1)\dots a(2m-1)=ds_{10}(3^m)/ds_{10}(3^{2m})\in [\frac1{9m},9m]$. But if $\lim a(m)\ne 1$, then $A(m)$ is either exponentially large ...
- 109k
4
votes
Accepted
Maximum product of digits of a perfect power
According to https://oeis.org/A052427,
"Pegg (1999) conjectured that the sequence of zeroless cubes (A052045) is finite. On April 19, 1999, Hickerson gave the counterexample: if $n \equiv2 \pmod ...
- 39.9k
3
votes
Accepted
Triangular repdigits
Following user523984's suggestion in the comments:
From triangle = repdigit $$\frac{k(k+1)}2 = \frac{d(10^j-1)}9$$ we get $$k = \frac{-1 \pm \frac13 \sqrt{9 + 8d(10^j-1)}}{2}$$ so we require $9 + 8d(...
- 7,226
3
votes
Does this sequence of ratios of digit sums have a limit?
Experimentally, the sequence converges to $1,$ at the logarithmic rate suggested in Fedor's answer. Here is the graph for the first 20000 numbers:
Now, when we fit the actual $ds_{10}(3^m),$ we get ...
- 96.4k
2
votes
Does a sequence of primes defined like this exists?
I agree that the sum of the base ten digits seems rather peripheral. However: given a positive integer $M;$ let $S_M$ be the set of primes $p$ so that the base $10$ digit sum of $Mp$ is a prime.
Q:...
- 30.1k
2
votes
Problem related to inequality of sum of digits of power sum
This is surely the case if $m\geq a-1$, when we have
$$S(a-1,m)<\int_1^a x^m\ dx<\frac{a^{m+1}}{m+1}\leq a^m.$$
- 34.3k
2
votes
Accepted
Partitioning integers into two parts and exploring relationships with positional numeral systems
Here is one kind of crazy example: Start with $A,B$ any two finite sets so that all the sums $a+b$ are distinct. $A=B=\emptyset$ for example. Now consider the integers in order $0,1,-1,2,-2,\cdots$ ...
- 30.1k
2
votes
What is the Mixed-Radix Numeral System of Best Radix Economy?
Since $\exp(3)$ is close to $20$, we can "score" close to the optimum using a cyclic pattern of base $2$ with base $5$.
Since the mlnatural logarithm of $2$ is closer to $1$ than tge natural ...
- 2,370
2
votes
Accepted
Consecutive prime numbers in permutations of digits of the first consecutive positive integers
here are some references to my previous research on a couple of strictly related open problems.
Here we show fascinating recurring patterns occurring in all the permutations of any element of the ...
- 1,451
1
vote
Golden ratio base
Not an answer but a long comment:
Such integer representations give rise to a nice language $\mathcal L$ in $\mathbb Z^*$ (finite words with letters in $\mathbb Z$). A finite word $w=x_1\ldots x_l$
...
- 17.9k
1
vote
Accepted
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
While it is very likely both numbers are absolutely normal, simply by appealing to the idea that there's no obvious reason why they should be abnormal, current proof techniques are very far from being ...
- 278
1
vote
The number of numbers no greater than n that are divisible by all their suffixes
Here's a start. If $y \cdot 10^d + z$ is such a number (with $y \in \{1,2,\ldots,9\}$ and $1 \le z < 10^d$), then $z \mid y \cdot 10^d$.
- 54.2k
1
vote
Is there a Bailey–Borwein–Plouffe (BBP) formula for e?
What you are actually looking for, is the Spigot Algorithm which is also known as "droplet algorithm".
That algorithm has been described a few years ago in the german "Spektrum der Wissenschaft" and ...
- 13.2k
Only top scored, non community-wiki answers of a minimum length are eligible