Piprimes
- Task
pi(n), the number of primes <= n, where pi(n) < 22
- Also see
-
- Prime-counting_function.
- Tables and hints  l by Tomás Oliveira e Silva.
- the OEIS entry: A0000720 pi(n), the number of primes <= n. Sometimes called PrimePi(n)....
ALGOL 68
<lang algol68>BEGIN # Show some values of pi(n) - the number of priems <= n #
# reurns a sieve of primes up to n #
PROC prime sieve = ( INT n )[]BOOL:
BEGIN
[ 1 : n ]BOOL p;
p[ 1 ] := FALSE; p[ 2 ] := TRUE;
FOR i FROM 3 BY 2 TO n DO p[ i ] := TRUE OD;
FOR i FROM 4 BY 2 TO n DO p[ i ] := FALSE OD;
FOR i FROM 3 BY 2 TO ENTIER sqrt( n ) DO
IF p[ i ] THEN FOR s FROM i * i BY i + i TO n DO p[ s ] := FALSE OD FI
OD;
p
END # prime sieve # ;
# show pi(n) for n up to 21 #
INT max number = 100; # guess of how large the primes we need are #
INT max pi = 21;
[]BOOL prime = prime sieve( max number );
INT pi := 0;
FOR i TO max number
WHILE IF prime[ i ] THEN pi +:= 1 FI;
pi <= max pi
DO
print( ( " ", whole( pi, -2 ) ) );
IF i MOD 10 = 0 THEN print( ( newline ) ) FI
OD
END</lang>
- Output:
0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10 11 11 11 11 11 11 12 12 12 12 13 13 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 16 17 17 18 18 18 18 18 18 19 19 19 19 20 20 21 21 21 21 21 21
Fermat
<lang fermat>n:=0; p:=0 while n<22 do !n;!' ';p:=p+1;if Isprime(p)=1 then n:=n+1; fi; od</lang>
- Output:
0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10 11 11 11 11 11 11 12 12 12 12 13 13 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 16 17 17 18 18 18 18 18 18 19 19 19 19 20 20 21 21 21 21 21 21
FreeBASIC
<lang freebasic>#define UPTO 22
- include "isprime.bas"
dim as integer running = 0, curr=0 do
curr += 1 if isprime(curr) then running += 1 if running = UPTO then exit do print running;" ";
loop print : end </lang>
- Output:
0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10 11 11 11 11 11 11 12 12 12 12 13 13 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 16 17 17 18 18 18 18 18 18 19 19 19 19 20 20 21 21 21 21 21 21
PARI/GP
<lang parigp>n=0; p=0 while(n<22, print(n); if(isprime(p),n=n+1);p=p+1)</lang>
Phix
integer ix = 1, n = 1, count = 0 sequence pi = {} while true do if get_prime(ix)<=n then count += 1 if count>=22 then exit end if ix += 1 end if n += 1 pi = append(pi,sprintf("%2d",count)) end while printf(1,"pi[1..%d]:\n%s\n",{length(pi),join_by(pi,1,10)})
- Output:
pi[1..78]: 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10 11 11 11 11 11 11 12 12 12 12 13 13 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 16 17 17 18 18 18 18 18 18 19 19 19 19 20 20 21 21 21 21 21 21
REXX
<lang rexx>/*REXX program finds and displays pi(n) for 0 < N ≤ prime(22) {the 22nd prime is 87},*/ /*────────────────────────── where the pi function returns the number of primes ≤ N.*/ parse arg hi cols . /*obtain optional argument from the CL.*/ if hi== | hi=="," then hi= 22 /* " " " " " " */ if cols== | cols=="," then cols= 10 /* " " " " " " */ call genP /*build array of semaphores for primes.*/ w= 10 /*width of a number in any column. */ @pips= ' number of primes that are (for all N) ≤ prime(22) which is ' commas(@.hi) if cols>0 then say ' index │'center(@pips, 1 + cols*(w+1) ) if cols>0 then say '───────┼'center("" , 1 + cols*(w+1), '─') idx= 1 /*initialize the index of output lines.*/ $=; pips= 0 /*a list of piPrimes numbers (so far). */
do j=1 for @.hi-1 /*gen list of piPrime numbers<prime(hi)*/
if !.j then pips= pips + 1 /*Is J prime? Then bump pips number.*/
if cols==0 then iterate /*Build the list (to be shown later)? */
c= commas(pips) /*maybe add commas to the number. */
$= $ right(c, max(w, length(c) ) ) /*add a Frobenius #──►list, allow big #*/
if j//cols\==0 then iterate /*have we populated a line of output? */
say center(idx, 7)'│' substr($, 2); $= /*display what we have so far (cols). */
idx= idx + cols /*bump the index count for the output*/
end /*j*/
if $\== then say center(idx, 7)"│" substr($, 2) /*possible display residual output.*/ say say 'Found ' commas(j-1)", the" @pips exit 0 /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ commas: parse arg ?; do jc=length(?)-3 to 1 by -3; ?=insert(',', ?, jc); end; return ? /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: !.= 0 /*placeholders for primes (semaphores).*/
@.1=2; @.2=3; @.3=5; @.4=7; @.5=11 /*define some low primes. */
!.2=1; !.3=1; !.5=1; !.7=1; !.11=1 /* " " " " flags. */
#=5; s.#= @.# **2 /*number of primes so far; prime². */
/* [↓] generate more primes ≤ high.*/
do j=@.#+2 by 2 until #>hi /*find odd primes from here on. */
parse var j -1 _; if _==5 then iterate /*J divisible by 5? (right dig)*/
if j// 3==0 then iterate /*" " " 3? */
if j// 7==0 then iterate /*" " " 7? */
/* [↑] the above 3 lines saves time.*/
do k=5 while s.k<=j /* [↓] divide by the known odd primes.*/
if j // @.k == 0 then iterate j /*Is J ÷ X? Then not prime. ___ */
end /*k*/ /* [↑] only process numbers ≤ √ J */
#= #+1; @.#= j; s.#= j*j; !.j= 1 /*bump # of Ps; assign next P; P²; P# */
end /*j*/; return</lang>
- output when using the default inputs:
index │ number of primes that are (for all N) ≤ prime(22) which is 79 ───────┼─────────────────────────────────────────────────────────────────────────────────────────────────────────────── 1 │ 0 1 2 2 3 3 4 4 4 4 11 │ 5 5 6 6 6 6 7 7 8 8 21 │ 8 8 9 9 9 9 9 9 10 10 31 │ 11 11 11 11 11 11 12 12 12 12 41 │ 13 13 14 14 14 14 15 15 15 15 51 │ 15 15 16 16 16 16 16 16 17 17 61 │ 18 18 18 18 18 18 19 19 19 19 71 │ 20 20 21 21 21 21 21 21 Found 78, the number of primes that are (for all N) ≤ prime(22) which is 79
Ring
<lang ring> load "stdlib.ring"
decimals(0) see "working..." + nl see "Piprimes are:" + nl
row = 0 limit1 = 400 Prim = []
for n = 1 to limit1
if isprime(n)
add(Prim,n)
ok
next
for n = 1 to len(Prim)
for m = 1 to len(Prim)
if Prim[m] > n
ind = m - 1
exit
ok
next
row = row + 1
see "" + ind + " "
if row%10 = 0
see nl
ok
next
see nl + "Found " + row + " Piprimes." + nl see "done..." + nl </lang>
- Output:
working... Piprimes are: 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10 11 11 11 11 11 11 12 12 12 12 13 13 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 16 17 17 18 18 18 18 18 18 19 19 19 19 20 20 21 21 21 21 21 21 Found 78 Piprimes. done...
Wren
<lang ecmascript>import "/math" for Int import "/seq" for Lst import "/fmt" for Fmt
var primes = Int.primeSieve(79) // go up to the 22nd var ix = 0 var n = 1 var count = 0 var pi = [] while (true) {
if (primes[ix] <= n) {
count = count + 1
if (count == 22) break
ix = ix + 1
}
n = n + 1
pi.add(count)
} System.print("pi(n), the number of primes <= n, where n >= 1 and pi(n) < 22:") for (chunk in Lst.chunks(pi, 10)) Fmt.print("$2d", chunk) System.print("\nHighest n for this range = %(pi.count).")</lang>
- Output:
pi(n), the number of primes <= n, where n >= 1 and pi(n) < 22: 0 1 2 2 3 3 4 4 4 4 5 5 6 6 6 6 7 7 8 8 8 8 9 9 9 9 9 9 10 10 11 11 11 11 11 11 12 12 12 12 13 13 14 14 14 14 15 15 15 15 15 15 16 16 16 16 16 16 17 17 18 18 18 18 18 18 19 19 19 19 20 20 21 21 21 21 21 21 Highest n for this range = 78.