# primes: Prime Numbers In pracma: Practical Numerical Math Functions

## 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) ```

### Example output

```  [1]   2   3   5   7  11  13  17  19  23  29  31  37  41  43  47  53  59  61
[19]  67  71  73  79  83  89  97 101 103 107 109 113 127 131 137 139 149 151
[37] 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251
[55] 257 263 269 271 277 281 283 293 307 311 313 317 331 337 347 349 353 359
[73] 367 373 379 383 389 397 401 409 419 421 431 433 439 443 449 457 461 463
[91] 467 479 487 491 499 503 509 521 523 541 547 557 563 569 571 577 587 593
[109] 599 601 607 613 617 619 631 641 643 647 653 659 661 673 677 683 691 701
[127] 709 719 727 733 739 743 751 757 761 769 773 787 797 809 811 821 823 827
[145] 829 839 853 857 859 863 877 881 883 887 907 911 919 929 937 941 947 953
[163] 967 971 977 983 991 997

Attaching package: 'gsl'

The following objects are masked from 'package:pracma':

erf, erfc, eta, expint_E1, expint_Ei, fact, hypot, psi, sinc, zeta

i         piN
[1,]  1           4           5 0.280109
[2,]  2          25          29 0.163239
[3,]  3         168         177 0.050979
[4,]  4        1229        1245 0.013094
[5,]  5        9592        9629 0.003833
[6,]  6       78498       78627 0.001637
[7,]  7      664579      664917 0.000509
[8,]  8     5761455     5762208 0.000131
[9,]  9    50847534    50849234 0.000033
[10,] 10   455052511   455055614 0.000007
[11,] 11  4118054813  4118066400 0.000003
[12,] 12 37607912018 37607950280 0.000001
```

pracma documentation built on Jan. 30, 2018, 3:01 a.m.