All Questions
29
questions
6
votes
0
answers
215
views
The sequence $0, 0, 1, 1, 3, 10, 52, 459, 1271, 10094, 63133,...$
Let $a_0$ be a permutation on $\{1, 2, ...,N\}$ (i.e. $a_0 \in S_N$) . For $n \geq 0$:
If $a_n(i+1) \geq a_n(i)$, then $a_{n+1}(i) = a_n(i+1) - a_n(i)$.
Otherwise, $a_{n+1}(i) = a_n(i+1) + a_n(i)$.
$...
- 1,029
0
votes
0
answers
34
views
Expected maximum occupation number for randomly distributed objects [duplicate]
Suppose you have $M$ distinguishable objects distributed amongst $N$ distinguishable boxes. Can you calculate the expected maximum occupation number $E_\text{max}(N,M)$? (in other words, the expected ...
- 2,339
3
votes
1
answer
338
views
Arrange numbers 1-12 around a circle so that any three consecutive numbers have a sum $\leq 20$?
My friend sent me the original question. Then he asked if 1-12 can be arranged in a way so that any three consecutive numbers have a sum that is not larger than 20. We guess the answer is no, since we ...
- 77
3
votes
2
answers
194
views
OEIS entry - A316312 has a question: Is it true that if k is a term then 100 * k is a term? [closed]
Refer https://oeis.org/A316312 - the sequence in OEIS.
The sequence says
Numbers n such that sum of the digits of the numbers 1, 2, 3, ... up
to (n - 1) is divisible by n.
A few terms from the ...
- 57
2
votes
0
answers
58
views
Which sets of $n-1$ non-multiples of $n$ can't make a multiple of $n$ using $+,-$?
This is a follow up to my previous question (see linked question).
In short, there it is shown that if $n$ is prime, then any set can make it.
I want to characterize sets $\mathbb A_n$ of multisets $...
- 12.5k
12
votes
2
answers
506
views
Integers less than $7000$ achievable by starting with $x=0$ and applying $x\to\lceil x^2/2\rceil$, $x\to\lfloor x/3\rfloor$, $x\to9x+2$
Problem
Robert is playing a game with numbers. If he has the number $x$, then in the next move, he can do one of the following:
Replace $x$ by $\lceil{\frac{x^2}{2}}\rceil$
Replace $x$ by $\lfloor{\...
- 707
7
votes
1
answer
154
views
For any $n-1$ non-zero elements of $\mathbb Z/n\mathbb Z$, we can make zero using $+,-$ if and only if $n$ is prime
Inspired by Just using $+$ ,$-$, $\times$ using any given n-1 integers, can we make a number divisible by n?, I wanted to first try to answer a simpler version of the problem, that considers only two ...
- 12.5k
0
votes
1
answer
118
views
Alcuin's problem of inheritance.
A certain father died and left as an inheritance to his three sons $30$ glass flasks, of which $10$ were full of oil, another $10$ were half full, while another $10$ were empty. Divide the oil and ...
1
vote
0
answers
216
views
A $10$ digit number with distinct digits such that the following holds:
A $10$ digit number with distinct digits is given and using all of its digits two new numbers are created. The sum of the two new numbers is $99999$ and their product is the same as the $10$ digit ...
- 12.7k
0
votes
3
answers
188
views
How to find a number $n$ such that $\frac{n}{\phi(n)} > 10$?
How to find a number $n$ such that $$\frac{n}{\phi(n)} > 10,$$ where $\phi(n)$ denotes the Euler's phi function?
I was trying to find the smallest one, so was keeping each prime once.
I tried with ...
- 12.7k
4
votes
1
answer
218
views
Longest consecutive runs of sums of $k$-subsets of first $n$ primes
Table of contents
[$1.$] Definition
[$2.$] Implication. (Motivation.)
[$3.$] Question. & Computed data.
[$4.$] Solutions of simplified variations.
[$5.$] Progress on solving the question.
[$6.$] ...
- 12.5k
4
votes
0
answers
176
views
Smallest number not expressible using first $n$ powers of $2$ (exactly once each), with $+$, $-$, $\times$, $\div$, and parentheses?
Motivation
Solution to this problem is a lower bound for a more general problem.
Problem
Given first $n$ powers of two: $1,2,4,8,16,\dots,2^{n-1}$ that all need to be used exactly once per number ...
- 12.5k
19
votes
0
answers
532
views
Largest consecutive integer using basic operations and optimal digits?
If you are first time reading this, you may want to read the summary section last.
Solution summary and questions
Sequence values
If the allowed operations are $(+,-,\times,\div)$ and parentheses $(...
- 12.5k
3
votes
0
answers
165
views
Generalization of Four Fours puzzle - optimal set of quadruplets?
Four fours is a math puzzle whose goal is to build numbers out of mathematical expressions using four fours, and a restricted set of mathematical operations and symbols.
Problem
I'm interested in ...
- 12.5k
4
votes
2
answers
401
views
Maximum run in binary digit expansions
For numbers between $2^{k-1}$ and $2^{k}-1$, how many have a maximum run of $n$ identical digits in base $2$? For instance, $1000110101111001$ in base $2$ has a maximum run of 4.
See picture below ...
- 3,645