All Questions
Tagged with integer-partitions prime-numbers
21
questions
1
vote
1
answer
46
views
Generating function of partitions of $n$ in $k$ prime parts.
I have been looking for the function that generates the partitions of $n$ into $k$ parts of prime numbers (let's call it $Pi_k(n)$). For example: $Pi_3(9)=2$, since $9=5+2+2$ and $9=3+3+3$.
I know ...
2
votes
1
answer
91
views
MacMahon partition function and prime detection (ref arXiv:2405.06451)
In the recent paper arXiv:2405.06451 the authors provide infinitely many characterizations of the primes using MacMahon partition functions: for $a>0$ the functions $M_a(n):=\sum\limits_{0<s_1&...
- 747
2
votes
1
answer
118
views
Does the partition function $p(n)$ generate infinite number of primes
Wikipedia says
As of June 2022, the largest known prime number among the values of $p(n)$ is $p(1289844341)$, with $40,000$ decimal digits
citing [1].
Is it known whether the partition function ...
- 3,341
1
vote
1
answer
282
views
Number of ways to write a positive integer as the sum of two coprime composites
I've recently learnt that every integer $n>210$ can be written as the sum of two coprime composites.
Similar to the totient function, is there any known function that works out the number of ways ...
- 103
2
votes
1
answer
74
views
Sum of Prime Factorizations and Primes
If I partition an integer and get the prime factorization of each partition, is there a way to tell if my original integer was a prime? For example, given the factorization of my partitions
$$71 = (56)...
- 157
2
votes
0
answers
45
views
Express number of partitions into prime numbers using partitions into natural numbers.
Let $P(n)$ is number of partitions of $n$ into natural numbers.
$R(n)$ is number of partitions of $n$ into prime numbers.
Is there any expression that relates $P(n)$ , and $R(n)$? I look for ...
- 1,382
0
votes
0
answers
51
views
Summation of a prime and a prime power
Is there an even number $n \in \mathbb{N}$ and two different primes $p,q<n$ which are not divisors of $n$, as well as $a,b \in \mathbb{N}$ with $a,b>1$, such that
$$
n=q+p^{a}=p+q^{b}
$$
? I ...
- 21
7
votes
1
answer
570
views
A surprising property of partitions into primes
I was studying some properties of partitions into primes and came across a surprising property. But before I talk about them, I am giving a definition.
Definition. A $k$-tuple $\lambda=(\lambda_1,\...
user840532
6
votes
2
answers
241
views
$p\equiv 1\pmod 4\Rightarrow p=a^2+b^2$ and $p\equiv 1\pmod 8\Rightarrow p=a^2+2b^2$, what about for $p\equiv 1\pmod {2^n}$ in general
Primes $p$ with $p\equiv 1\pmod 4$ can be written as $p=a^2+b^2$ for some integers $a,b$. For $p\equiv 1\pmod 8$ we have $p=a^2+2b^2$. Can primes that satisfy $p\equiv 1\pmod{2^n}$ for $n>3$ be ...
- 1,940
7
votes
1
answer
190
views
Finding $z=x+y$ such that $x^2 + y^2$ is prime
For which integers $z$ can one write $z=x+y$ such that $x^2+y^2$ is prime?
It feels like it should be possible for all odd $z>1$, and I have tried to adapt Euler's proof of Girard/Fermat's ...
- 71
6
votes
0
answers
181
views
Almost a prime number recurrence relation
For the number of partitions of n into prime parts $a(n)$ it holds
$$a(n)=\frac{1}{n}\sum_{k=1}^n q(k)a(n-k)\tag 1$$
where $q(n)$ the sum of all different prime factors of $n$.
Due to https://oeis....
- 13.9k
0
votes
1
answer
53
views
Question about the partitions of a natural number
There is a function that counts the number of partitions of with $n$ digits?
I am aware of the partition function studied by Ramanujan, but what I want is a subset of the partitions that are counted ...
- 2,505
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
0
votes
0
answers
40
views
The number of $n\in\mathbb{N}$ with $p(n)$ is prime
Let $p(n)$ denote the number of integer partitions of $n$ for $n\in\mathbb{N}$.
Is it possible to list the cases where $p(n)$ is prime? Are such natural numbers finite (if so how to compute a bound) ...
- 4,852
4
votes
0
answers
218
views
Relationship between Riemann Zeta function and Prime zeta function
In his paper, Daniel Grunberg shows a relationship between the Stirling Numbers of the first kind and the Harmonic numbers via series of partitions (see Equation 3.1 on Page 5 in the link above). If ...
- 3,557