All Questions
Tagged with integer-partitions elementary-number-theory
76
questions
4
votes
1
answer
243
views
How to count all the solutions for $\sum\limits_{i=1}^{n} \frac{1}{2^{k_i}}= 1$ for $k_i\in \Bbb{N}$ and $n$ a fixed positive integer?
After reading this question, I would like to just count all solutions for:
$$\frac{1}{2^{k_1}} + \frac{1}{2^{k_2}} + \frac{1}{2^{k_3}} + \dots + \frac{1}{2^{k_n}}=1$$
for $k_i\in \Bbb{N}$ (we can ...
- 2,103
1
vote
1
answer
243
views
Is it possible to use partitions of an odd integer to generate primes in a given interval?
We start with the partition of $N=5$.
$$5$$
$$4+1$$
$$3+2$$
$$3+1+1$$
$$2+2+1$$
$$2+1+1+1$$
$$1+1+1+1+1$$
Then we form the sum of squares (no limit on the number of elements) to get:
$$4^2+1^2=17$$
...
- 1,058
2
votes
1
answer
77
views
Maximizing $\sum\left(\lfloor \frac{n_i}{2} \rfloor+1\right)$ for a partition $\{n_i\}$ of $N$
Let $N$ be a natural number and $\{n_i\}$ be a partition of $N$; by this we mean $1\leq i\leq k$ for some natural number $k$ and $N=n_1+n_2+\cdots+n_k$ where $n_1\geq n_2\geq\ldots\geq n_k\geq1$.
For ...
- 1,107
2
votes
3
answers
249
views
How to count all restricted partitions of the number $155$ into a sum of $10$ natural numbers between $[0,30]$? [closed]
How to count all restricted partitions of the number $155$ into a sum of $10$ natural numbers between $[0,30]$?
I really have no clue what to do with this one. Thanks for any help!
- 29
0
votes
0
answers
181
views
A generating function $G(x)=-\frac{\frac{1}{x^5}(1+\frac{1}{x})(1-\frac{1}{x^2})}{((1-\frac{1}{x})(1-\frac{1}{x^3}))^2}$ related to partitions of $6n$
Fix a sequence $a_n={n+2\choose 2}$ of triangular numbers with the initial condition $a_0=1$,such that
$1,3,6,10,15,21,\dots$
given by
$F(x)=\frac{1}{(1-x)^3}=\sum_{n=0}^{\infty} a_n x^n\tag1$
...
- 2,813
0
votes
1
answer
33
views
Converse of Ramanujan's Congruences
Of Ramanujan's famous congruences for the partition function, $p(5k+4)\equiv0\mod 5$, $p(7k+5)\equiv0\mod7$, and so on, does the converse also hold? For example, if $p(n)\equiv0\mod5$, does that mean $...
- 101
0
votes
1
answer
43
views
Equation to approximate a Partition-like function
The partition function for $n$, $P(n)$ gives the number of partitions that exist for $n$.
I've been trying to find a function that gives the number of partitions where order matters, e.g. $1+2+3$ is ...
- 109
2
votes
0
answers
133
views
Equal and unequal partition?
Can someone please tell me what equal and unequal partition is? And if the question, I'm just putting an example here, is asking you to find at least 3 equal and unequal partitions of 2020 into 4 ...
- 21
5
votes
1
answer
231
views
A Conjectured Mathematical Constant For Base-10 Normal Numbers.
Question 1: Let $a$ be a real number with a base-10 decimal
representation $a_1a_2\ldots a_n \ldots$ Denote the number of ways to
write $a_n$ as the sum of positive integers as $p(a_n)$ - also ...
- 3,758
9
votes
2
answers
264
views
P-graph of partition elements of 100 under common divisibility relation
Given a multiset of positive integers, its P-graph is the loopless graph whose vertex set consists of those integers, any two of which are joined by an edge if they have a common divisor greater than ...
- 1,162
10
votes
3
answers
3k
views
Number of partitions of $50$
Does someone know the number of partitions of the integer $50$? I mean, in how many ways can I write $50$ as a sum of positive integers?
I know that there's a table by Euler, which is useful to know ...
- 315
2
votes
0
answers
69
views
Is there accepted notation for the set of ways of partitioning a natural number $a$ into $b$ parts?
Recall the following:
Multinomial Theorem. For all finite sets $X$, we have:
$$\left(\sum_{x \in X} x\right)^n = \sum_{a}[a](X)^a$$
where $a$ ranges over the set of partitions of $n$ into ...
- 68.1k
1
vote
1
answer
241
views
Elementary proof of: Any integer is a sum of distinct numbers in {1,2,3,5,7,11,13,17,...}
Let $\mathbb P^1=\{1\}\cup\mathbb P$, the set of positive non composites. I have reason to believe that it is proved that all numbers greater than $6$ is a sum of distinct primes, and hence all $n\in\...
- 13.9k
2
votes
0
answers
719
views
$4\sum_{m,n=1}^{\infty}\frac{q^{n+m}}{(1+q^n)(1+q^m)}(z^{n-m}+z^{m-n})=8\sum_{m,n=1}^{\infty}\frac{q^{n+2m}}{(1+q^n)(1+q^{n+m})}(z^m+z^{-m})$
To prove the identity $$4\sum_{m,n=1}^{\infty}\frac{q^{n+m}}{(1+q^n)(1+q^m)}(z^{n-m}+z^{m-n})=8\sum_{m,n=1}^{\infty}\frac{q^{n+2m}}{(1+q^n)(1+q^{n+m})}(z^m+z^{-m})$$ I replaced $m-n$ by $k$ in LHS ...
1
vote
1
answer
133
views
Partitions of 2017 natural numbers [closed]
Suppose we have 2017 natural numbers, such that each 2016 can be grouped into 2 groups with equal sum and equal number of elements. Prove that all numbers are equal.
- 193