# Tagged Questions

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### How was the $3x+1$ problem checked up to $5 \times 2^{60}$?

The Wikipedia article for the Collatz conjecture states that: The conjecture has been checked by computer for all starting values up to $5 \times 2^{60} \approx 5.764 \times 10^{18}$. It gives ...
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### Efficient computation of $\sum_{k=1}^n \lfloor \frac{n}{k}\rfloor$

I realize there is probably not a closed form, but is there an efficient way to calculate the following expression? $$\sum_{k=1}^n \left\lfloor \frac{n}{k}\right\rfloor$$ I've noticed \sum_{k=1}^n ...
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### How are 10-20 digit multiperfect and hemiperfect numbers efficiently computed?

This numericana item on multiperfect and hemiperfect numbers contains some impressively enormous numbers. How were these actually computed ? The associated OEIS pages (A007691 & A159907) just ...
Def: a pair natural numbers $a$, $b$, $a\ne b$ are an Amicable pair if $\sum_{d|a,a\ne d}d = b$ and $\sum_{d|b, b\ne d}d = a$. Ok. So I'm trying to optimize a calculation for finding the number of ...