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### Counting number of primorial factors

Denote $$P(n)=\prod_{p\in\mathsf{Primes}\leq n}p$$ signifying $n^{\mbox{th}}$ primorial.
We know that $P(n)$ has approximately $n/\log2$ bits ...

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### In which orders can the numbers of prime factors of consecutive integers be?

Let $\omega(m)$ be the number of distinct prime divisors of a positive integer $m>1$. I am interested in the relative orders in which the numbers $\omega(n+1),...,\omega(n+k)$ can occur.
Given ...

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### Reference request: Darboux properties of real-valued set functions (measures, densities, etc.)

Fix a set $S$ and let $f: \mathcal P(S) \rightharpoonup \mathbf R$ be a real-valued partial function on the power set of $S$; denote by $\mathcal D$ the domain of $f$. We say that $f$ has:
(i) the ...

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### Probability that an integer contains no $1\bmod 4$ prime factor

$n$ represents integer variable.
What is the probability that and integer contains at most $r(n)$ prime factors of form $1\bmod 4$ where $r(n)$ is a function of $\omega(n)$ (number of distinct prime ...

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252 views

### A variant frobenius problem

From Sylvester's theorem we know that using only coins of sizes $a,b$, we can change exactly $\frac{(a-1)(b-1)}2$ different big coins up to $(a-1)(b-1)$.
Denote sets ...

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### On collision-free sets of residues of integers

Denote $\pi_m$ to be collection of primes in $[2^{m},2^{m+1}]$.
Denote $\psi_{n,m}$ to be collection of integers of form $ab$ where $a\in\pi_n$, $b\in\pi_m$.
Given $n,t\in\Bbb N$, what is the ...

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129 views

### Relation between the densities of two functions and their additive convolution?

Suppose that for two integral-valued arithmetic functions $f_i$ ($i=1,2$), the following values are known:
$$ \lim_{n\to \infty} \frac{ \{n:f_i(n) \text{ is odd}\}\cap \{0,1,\ldots, n-1\} }{n}.$$
(In ...