primeCount {RcppAlgos}

Prime Counting Function \pi(x)

Description

Prime counting function for counting the prime numbers less than an integer, n, using Legendre's formula. It is based on the the algorithm developed by Kim Walisch found here: kimwalisch/primecount.

Usage

primeCount(n, nThreads = NULL)

Arguments

n

Positive number

nThreads

Specific number of threads to be used. The default is NULL.

Details

Legendre's Formula for counting the number of primes less than n makes use of the inclusion-exclusion principle to avoid explicitly counting every prime up to n. It is given by:

\pi(x) = \pi(\sqrt x) + \Phi(x, \sqrt x) - 1

Where \Phi(x, a) is the number of positive integers less than or equal to x that are relatively prime to the first a primes (i.e. not divisible by any of the first a primes). It is given by the recurrence relation (p_a is the ath prime (e.g. p_4 = 7)):

\Phi(x, a) = \Phi(x, a - 1) + \Phi(x / p_a, a - 1)

This algorithm implements five modifications developed by Kim Walisch for calculating \Phi(x, a) efficiently.

1. Cache results of \Phi(x, a)

2. Calculate \Phi(x, a) using \Phi(x, a) = (x / pp) * \phi(pp) + \Phi(x mod pp, a) if a <= 6

   • pp = 2 * 3 * ... * prime[a]

   • \phi(pp) = (2 - 1) * (3 - 1) * ... * (prime[a] - 1) (i.e. Euler's totient function)

3. Calculate \Phi(x, a) using \pi(x) lookup table

4. Calculate all \Phi(x, a) = 1 upfront

5. Stop recursion at 6 if \sqrt x >= 13 or \pi(\sqrt x) instead of 1

Value

Whole number representing the number of prime numbers less than or equal to n.

Note

The maximum value of n is 2^{53} - 1

Author(s)

Joseph Wood

References

• Computing \pi(x): the combinatorial method

  • Tomás Oliveira e Silva, Computing pi(x): the combinatorial method, Revista do DETUA, vol. 4, no. 6, March 2006, p. 761. https://sweet.ua.pt/tos/bib/5.4.pdf

• 53-bit significand precision

See Also

primeSieve

Examples

## Get the number of primes less than a billion
primeCount(10^9)

## Using nThreads
system.time(primeCount(10^10, nThreads = 2))