# primes: Prime Numbers

### Description

Generate a list of prime numbers less or equal `n`, resp. between `n1` and `n2`.

### Usage

 `1` ``` primes(n) ```

### Arguments

 `n` nonnegative integer greater than 1.

### Details

The list of prime numbers up to `n` is generated using the "sieve of Erasthostenes". This approach is reasonably fast, but may require a lot of main memory when `n` is large.

In double precision arithmetic integers are represented exactly only up to 2^53 - 1, therefore this is the maximal allowed value.

### Value

vector of integers representing prime numbers

`isprime, factors`

### Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32``` ```primes(1000) ## Not run: ## Appendix: Logarithmic Integrals and Prime Numbers (C.F.Gauss, 1846) library('gsl') # 'European' form of the logarithmic integral Li <- function(x) expint_Ei(log(x)) - expint_Ei(log(2)) # No. of primes and logarithmic integral for 10^i, i=1..12 i <- 1:12; N <- 10^i # piN <- numeric(12) # for (i in 1:12) piN[i] <- length(primes(10^i)) piN <- c(4, 25, 168, 1229, 9592, 78498, 664579, 5761455, 50847534, 455052511, 4118054813, 37607912018) cbind(i, piN, round(Li(N)), round((Li(N)-piN)/piN, 6)) # i pi(10^i) Li(10^i) rel.err # -------------------------------------- # 1 4 5 0.280109 # 2 25 29 0.163239 # 3 168 177 0.050979 # 4 1229 1245 0.013094 # 5 9592 9629 0.003833 # 6 78498 78627 0.001637 # 7 664579 664917 0.000509 # 8 5761455 5762208 0.000131 # 9 50847534 50849234 0.000033 # 10 455052511 455055614 0.000007 # 11 4118054813 4118066400 0.000003 # 12 37607912018 37607950280 0.000001 # -------------------------------------- ## End(Not run) ```

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