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Results tagged with divisors-multiples
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user 9232
For questions on divisors and multiples, mainly but not exclusively of integers, and related and derived notions such as sums of divisors, perfect numbers and so on.
1
vote
A truncated divisor function sum
The $n=ab$ trick is very effective. There is a similar trick of writing in a unique way $n=ab$ with $n$ squarefree, $b$ a square and I was wondering whether this could also work here.
- 3,062
1
vote
Estimating $\sum_{n\leq x: n \in A} d(n)^a$ from below for large sets $A\subset \{1,2,\ldots...
There is a cheap way via Cauchy's inequality in case you do no want to use Erdos--Kac. I will only do it for $a=1$ just to outline the idea.
$$\frac{x}{2} (1+o(1) ) \leq
\sum_{n \leq x } 1_A(x)=
\su …
- 3,062
14
votes
3
answers
1k
views
On the number of consecutive divisors of an integer
Define for $n \in \mathbb{N}$ the function $$\tau_1(n):=\sum_{\substack{d|n, \\ d+1|n}}1,$$ i.e. the number of consecutive divisors of an integer. The average of $\tau_1(n)$ is $1$ since $$\sum_{n\leq …
- 3,062