Question: Which of the following are reduced modulo residue systems modulo 18?
$a. 1,5,25,125,625,3125$
$b. 5, 11, 17, 23, 29, 35$
$c. 1, 25, 49, 121, 169, 289$
$d. 1, 5, 7, 11, 13, 17$
Attempt: Our mod n is mod 18, so n = 18. Now, we need to remove the integers that aren't prime to n. Now, ...
In mathematics, Stirling's approximation (or Stirling's formula) is an approximation for factorials. It is a very powerful approximation, leading to accurate results for even small values of n. It is named after James Stirling.
The formula as typically used in applications is
The next term in the O(ln(n)) is (1/2)ln(2πn); a more precise variant of the formula is therefore
Being an asymptotic formula, Stirling's approximation has the property that
Sometimes, bounds for rather than asymptotics are required: one has, for all
so for all the ratio is always between A019727 and A001113.
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5
In the chatroom we discussed about the asymptotic of $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}$, and if we think of the inverse tangent integral, it's easy to see that $\displaystyle \sum_{i = 1}^n \sum_{j = 1}^n \frac1{i^2+j^2}\approx \operatorname{Ti}_2(n)\approx \frac{\pi}{2...
