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1
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Natural density of integers such that $\forall 1\leq k\leq n-1$ satisfy $\operatorname{rad}(k)+\operatorname{rad}(n-k)\geq \operatorname{rad}(n)$
For an integer $n>1$ we defined its square-free kernel $$\operatorname{rad}(n)=\prod_{\substack{p\mid n\\p \text{ prime}}}p$$
as the product of distinct prime factors dividing it, with the ...
user243301
2
votes
0
answers
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Compare $\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\rfloor\right)}$ and $\sum_{k\mid n}k^{\operatorname{rad}\left(\frac{n}{k}\right)}$
I would like to know how do a comparison between the sizes of these functions defined for integers $n\geq 1$, when $n$ is large
$$f(n):=\sum_{k=1}^n k^{\operatorname{rad}\left(\lfloor\frac{n}{k}\...
user243301
2
votes
1
answer
203
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Do we know the rate of divergence of the sum of reciprocals of the $k$-almost primes?
A $k$-almost prime is a positive integer having exactly $k$ prime factors, not necessarily distinct. Let $\mathbb{P}_k$ be the set of the $k$-almost primes and let $$ \rho_k(n):=\sum\limits_{\substack{...
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