Divisibility

Dec 2015
1,078
166
Earth
How many numbers are divisible by \(\displaystyle {1,2,..,n}\) inside interval \(\displaystyle [1,2n]\) ?
 
Jun 2014
650
54
USA
So for any $n \in \mathbb{N}$, you’re asking for the cardinality of $\{ x \in \{1, 2, 3, \dots, 2n\} : \frac{2n}{x} \in \mathbb{N} \}$, correct?
 
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Jun 2019
493
262
USA
Interpreting the question the same way as AplanisTophet, I'm pretty sure the answer is no more than two.

[1, 2] are both divisible by 1.
[2, 4] $\in$ [1..4] are divisible by 1 and 2.
[6] $\in$ [1..6] is divisible by 1, 2, and 3.
No values in [1..8] are divisible by 1, 2, 3, and 4. (LCM 12)
No values in [1..10] are divisible by 1, 2, 3, 4, and 5. (LCM 60)
No values in [1..12] are divisible by 1, 2, 3, 4, 5, and 6. (LCM 60)
No values in [1..14] are divisible by {1..7}. (LCM 420)
No values in [1..16] are divisible by {1..8}. (LCM 840)
&c. &c.
 
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