Tagged Questions
2
votes
1answer
68 views
Related to the partition of $n$ where $p(n,2)$ denotes the number of partitions $\geq 2$
I was trying the following question from the Number theory book of Zuckerman.
If $p(n,2)$ denotes the number of partition of n with parts $\geq2$, prove that $p(n,2)>p(n-1,2)$ for all $n\geq 8$, ...
4
votes
0answers
88 views
For every partition $\pi$ of a fixed integer $n$, $\sum{F(\pi)}=\sum{G(\pi)}$
I need to prove the following question.
For every partition $\pi$ of a fixed integer $n$, define $F(\pi)$=number of occurrences of 1 as a summand, and $G(\pi)$=no. of distinct summands in the ...
2
votes
0answers
91 views
Partition minimizing maximum of Euler's totient function across terms
Given natural numbers $M$ and $N$, I'd like to find a partition of $2^N$ with $M$ or fewer terms, $t_1 + t_2 + ... + t_M$, such that $\max(\phi(t_1), \phi(t_2), ..., \phi(t_M))$ is minimized, where ...
0
votes
2answers
63 views
Combination Question
I currently have an open question about counting the possible ways of summing numbers. I am still exploring all the ideas provided - those within my level of understanding. This is a question ...
4
votes
2answers
177 views
Find all ways to factor a number
An example of what I'm looking for will probably explain the question best. 24 can be written as:
12 · 2
6 · 2 · 2
3 · 2 · 2 · 2
8 · 3
4 · 2 · 3
6 · 4
I'm familiar with finding all the prime ...
3
votes
1answer
161 views
Lower bounds for the partition function
In this question we consider the partition function $p(n)$ - that is, the number of ways to express $n$ as a sum of positive integers.
One easy exercise is to show that $$ p(n) \geq 2^{\lfloor ...
8
votes
4answers
603 views
Prime Partition
A prime partition of a number is a set of primes that sum to the number. For instance, {2 3 7} is a prime partition of $12$ because $2 + 3 + 7 = 12$. In fact, there ...
1
vote
1answer
165 views
The coin change problem in the quantitative way
Today,I came across this problem:
Suppose you have a currency, named miso, in three denominations, $1,
10$ and $50$. In how many ways can $107$ miso be given in this
currency if you have ...
1
vote
1answer
298 views
Number of solutions of Frobenius equation
I have one problem which needs to count the number of solution of the equation $$2x+7y+11z=42$$
where $x,y,z \in \{0,1,2,3,4,5,\dots\}$.
My attempt:
I noticed that that maximum value of $z$ could ...
4
votes
2answers
263 views
Natural set to express any natural number as sum of two in the set
Any natural number can be expressed as the sum of three triangular numbers, or as four square numbers. The natural analog for expressing numbers as the sum of two others would apparently be the sum ...
