Is there a way to find the approximate value of the nth prime?

Question

Is there a function which will return the approximate value of the n th prime? I think this would be something like an approximate inverse prime counting function.

Answer 1

Thanks for all of those answers. I suspected there was something fairly simple like that, but I couldn't find it at the time. I've done a bit more research too.

As I want it for a sieve to generate the first n prime numbers, I want the approximation to be greater than or equal to the nth prime. (Therefore, I want the upper bound of the nth prime number.)

Wikipedia gives the following upper bound for n >= 6

p_n <= n log n + n log log n   (1)

where p_n is the nth prime, and log is the natural logarithm. This is a good start, but it can overestimate by a not inconsiderable amount. This article in The College Mathematics Journal gives a tighter upper bound for n >= 7022

p_n <= n log n + n (log log n - 0.9385)   (2)

This is a much tighter bound as the following table shows

n         p_n         approx 1    error%  approx 2    error%
1         2                            
10        29          31          6.90 
100       541         613         13.31
1000      7919        8840        11.63
10000     104729      114306      9.14    104921      0.18
100000    1299709     1395639     7.38    1301789     0.16
1000000   15485863    16441302    6.17    15502802    0.11
10000000  179424673   188980382   5.33    179595382   0.10

I implemented my nth prime approximation function to use the second approximation for n >= 7022, the first approximation for 6 <= n < 7022, and an array lookup for the first 5 prime numbers.

(Although the first method isn't a very tight bound, especially for the range where I use it, I am not concerned as I want this for a sieve, and a sieve of smaller numbers is computationally cheap.)