Strong and weak primes

From Rosetta Code
Task
Strong and weak primes
You are encouraged to solve this task according to the task description, using any language you may know.


Definitions   (as per number theory)
  •   The   prime(p)   is the   pth   prime.
  •   prime(1)   is   2
  •   prime(4)   is   7
  •   A   strong   prime   is when     prime(p)   is   >   [prime(p-1) + prime(p+1)] ÷ 2
  •   A     weak   prime   is when     prime(p)   is   <   [prime(p-1) + prime(p+1)] ÷ 2


Note that the definition for   strong primes   is different when used in the context of   cryptography.


Task
  •   Find and display (on one line) the first   36   strong primes.
  •   Find and display the   count   of the strong primes below   1,000,000.
  •   Find and display the   count   of the strong primes below 10,000,000.
  •   Find and display (on one line) the first   37   weak primes.
  •   Find and display the   count   of the weak primes below   1,000,000.
  •   Find and display the   count   of the weak primes below 10,000,000.
  •   (Optional)   display the   counts   and   "below numbers"   with commas.

Show all output here.


Related Task


Also see



ALGOL 68[edit]

Works with: ALGOL 68G version Any - tested with release 2.8.3.win32
# find and count strong and weak primes                                       #
PR heap=128M PR # set heap memory size for Algol 68G #
# returns a string representation of n with commas #
PROC commatise = ( INT n )STRING:
BEGIN
STRING result := "";
STRING unformatted = whole( n, 0 );
INT ch count := 0;
FOR c FROM UPB unformatted BY -1 TO LWB unformatted DO
IF ch count <= 2 THEN ch count +:= 1
ELSE ch count := 1; "," +=: result
FI;
unformatted[ c ] +=: result
OD;
result
END # commatise # ;
# sieve values #
CHAR prime = "P"; # unclassified/average prime #
CHAR strong = "S"; # strong prime #
CHAR weak = "W"; # weak prime #
CHAR composite = "C"; # non-prime #
# sieve of Eratosthenes: sets s[i] to prime if i is a prime, #
# composite otherwise #
PROC sieve = ( REF[]CHAR s )VOID:
BEGIN
# start with everything flagged as prime #
FOR i TO UPB s DO s[ i ] := prime OD;
# sieve out the non-primes #
s[ 1 ] := composite;
FOR i FROM 2 TO ENTIER sqrt( UPB s ) DO
IF s[ i ] = prime THEN FOR p FROM i * i BY i TO UPB s DO s[ p ] := composite OD FI
OD
END # sieve # ;
 
INT max number = 10 000 000;
# construct a sieve of primes up to slightly more than the maximum number #
# required for the task, as we may need an extra prime for the classification #
[ 1 : max number + 1 000 ]CHAR primes;
sieve( primes );
# classify the primes #
# find the first three primes #
INT prev prime := 0;
INT curr prime := 0;
INT next prime := 0;
FOR p FROM 2 WHILE prev prime = 0 DO
IF primes[ p ] = prime THEN
prev prime := curr prime;
curr prime := next prime;
next prime := p
FI
OD;
# 2 is the only even prime so the first three primes are the only case where #
# the average of prev prime and next prime is not an integer #
IF REAL avg = ( prev prime + next prime ) / 2;
curr prime > avg THEN primes[ curr prime ] := strong
ELIF curr prime < avg THEN primes[ curr prime ] := weak
FI;
# classify the rest of the primes #
FOR p FROM next prime + 1 WHILE curr prime <= max number DO
IF primes[ p ] = prime THEN
prev prime := curr prime;
curr prime := next prime;
next prime := p;
IF INT avg = ( prev prime + next prime ) OVER 2;
curr prime > avg THEN primes[ curr prime ] := strong
ELIF curr prime < avg THEN primes[ curr prime ] := weak
FI
FI
OD;
INT strong1 := 0, strong10 := 0;
INT weak1 := 0, weak10 := 0;
FOR p WHILE p < 10 000 000 DO
IF primes[ p ] = strong THEN
strong10 +:= 1;
IF p < 1 000 000 THEN strong1 +:= 1 FI
ELIF primes[ p ] = weak THEN
weak10 +:= 1;
IF p < 1 000 000 THEN weak1 +:= 1 FI
FI
OD;
INT strong count := 0;
print( ( "first 36 strong primes:", newline ) );
FOR p WHILE strong count < 36 DO IF primes[ p ] = strong THEN print( ( " ", whole( p, 0 ) ) ); strong count +:= 1 FI OD;
print( ( newline ) );
print( ( "strong primes below 1,000,000: ", commatise( strong1 ), newline ) );
print( ( "strong primes below 10,000,000: ", commatise( strong10 ), newline ) );
print( ( "first 37 weak primes:", newline ) );
INT weak count := 0;
FOR p WHILE weak count < 37 DO IF primes[ p ] = weak THEN print( ( " ", whole( p, 0 ) ) ); weak count +:= 1 FI OD;
print( ( newline ) );
print( ( " weak primes below 1,000,000: ", commatise( weak1 ), newline ) );
print( ( " weak primes below 10,000,000: ", commatise( weak10 ), newline ) )
Output:
first 36 strong primes:
 11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439
strong primes below   1,000,000: 37,723
strong primes below  10,000,000: 320,991
first 37   weak primes:
 3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401
  weak primes below   1,000,000: 37,780
  weak primes below  10,000,000: 321,750

Go[edit]

package main
 
import "fmt"
 
func sieve(limit int) []bool {
limit++
// True denotes composite, false denotes prime.
// Don't bother marking even numbers >= 4 as composite.
c := make([]bool, limit)
c[0] = true
c[1] = true
 
p := 3 // start from 3
for {
p2 := p * p
if p2 >= limit {
break
}
for i := p2; i < limit; i += 2 * p {
c[i] = true
}
for {
p += 2
if !c[p] {
break
}
}
}
return c
}
 
func commatize(n int) string {
s := fmt.Sprintf("%d", n)
le := len(s)
for i := le - 3; i >= 1; i -= 3 {
s = s[0:i] + "," + s[i:]
}
return s
}
 
func main() {
// sieve up to 10,000,019 - the first prime after 10 million
const limit = 1e7 + 19
sieved := sieve(limit)
// extract primes
var primes = []int{2}
for i := 3; i <= limit; i += 2 {
if !sieved[i] {
primes = append(primes, i)
}
}
// extract strong and weak primes
var strong []int
var weak = []int{3} // so can use integer division for rest
for i := 2; i < len(primes)-1; i++ { // start from 5
if primes[i] > (primes[i-1]+primes[i+1])/2 {
strong = append(strong, primes[i])
} else if primes[i] < (primes[i-1]+primes[i+1])/2 {
weak = append(weak, primes[i])
}
}
 
fmt.Println("The first 36 strong primes are:")
fmt.Println(strong[:36])
count := 0
for _, p := range strong {
if p >= 1e6 {
break
}
count++
}
fmt.Println("\nThe number of strong primes below 1,000,000 is", commatize(count))
fmt.Println("\nThe number of strong primes below 10,000,000 is", commatize(len(strong)))
 
fmt.Println("\nThe first 37 weak primes are:")
fmt.Println(weak[:37])
count = 0
for _, p := range weak {
if p >= 1e6 {
break
}
count++
}
fmt.Println("\nThe number of weak primes below 1,000,000 is", commatize(count))
fmt.Println("\nThe number of weak primes below 10,000,000 is", commatize(len(weak)))
}
Output:
The first 36 strong primes are:
[11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439]

The number of strong primes below 1,000,000 is 37,723

The number of strong primes below 10,000,000 is 320,991

The first 37 weak primes are:
[3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401]

The number of weak primes below 1,000,000 is 37,780

The number of weak primes below 10,000,000 is 321,750

Lua[edit]

This could be made faster but favours readability. It runs in about 3.3 seconds in LuaJIT on a 2.8 GHz core.

-- Return a table of the primes up to n, then one more
function primeList (n)
local function isPrime (x)
for d = 3, math.sqrt(x), 2 do
if x % d == 0 then return false end
end
return true
end
local pTable, j = {2, 3}
for i = 5, n, 2 do
if isPrime(i) then
table.insert(pTable, i)
end
j = i
end
repeat j = j + 2 until isPrime(j)
table.insert(pTable, j)
return pTable
end
 
-- Return a boolean indicating whether prime p is strong
function isStrong (p)
if p == 1 or p == #prime then return false end
return prime[p] > (prime[p-1] + prime[p+1]) / 2
end
 
-- Return a boolean indicating whether prime p is weak
function isWeak (p)
if p == 1 or p == #prime then return false end
return prime[p] < (prime[p-1] + prime[p+1]) / 2
end
 
-- Main procedure
prime = primeList(1e7)
local strong, weak, sCount, wCount = {}, {}, 0, 0
for k, v in pairs(prime) do
if isStrong(k) then
table.insert(strong, v)
if v < 1e6 then sCount = sCount + 1 end
end
if isWeak(k) then
table.insert(weak, v)
if v < 1e6 then wCount = wCount + 1 end
end
end
print("The first 36 strong primes are:")
for i = 1, 36 do io.write(strong[i] .. " ") end
print("\n\nThere are " .. sCount .. " strong primes below one million.")
print("\nThere are " .. #strong .. " strong primes below ten million.")
print("\nThe first 37 weak primes are:")
for i = 1, 37 do io.write(weak[i] .. " ") end
print("\n\nThere are " .. wCount .. " weak primes below one million.")
print("\nThere are " .. #weak .. " weak primes below ten million.")
Output:
The first 36 strong primes are:
11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439

There are 37723 strong primes below one million.

There are 320991 strong primes below ten million.

The first 37 weak primes are:
3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401

There are 37780 weak primes below one million.

There are 321750 weak primes below ten million.

Pascal[edit]

Converting the primes into deltaPrime, so that its easy to check the strong- /weakness. Startprime 2 +1 -> (3)+2-> (5)+2 ->(7) +4-> (11)+2 .... 1,2,2,4,2,4,2,4,6,2,.... By using only odd primes startprime is 3 and delta -> delta/2

If deltaAfter < deltaBefore than a strong prime is found.

program WeakPrim;
{$IFNDEF FPC}
{$AppType CONSOLE}
{$ENDIF}
const
PrimeLimit = 1000*1000*1000;//must be >= 2*3;
type
tLimit = 0..(PrimeLimit-1) DIV 2;
tPrimCnt = 0..51*1000*1000;
tWeakStrong = record
strong,
balanced,
weak : NativeUint;
end;
var
primes: array [tLimit] of byte; //always initialized with 0 at startup
delta : array [tPrimCnt] of byte;
cntWS : tWeakStrong;
deltaCnt :NativeUint;
 
procedure sieveprimes;
//Only odd numbers, minimal count of strikes
var
spIdx,sieveprime,sievePos,fact :NativeUInt;
begin
spIdx := 1;
repeat
if primes[spIdx]=0 then
begin
sieveprime := 2*spIdx+1;
fact := PrimeLimit DIV sieveprime;
if Not(odd(fact)) then
dec(fact);
IF fact < sieveprime then
BREAK;
sievePos := ((fact*sieveprime)-1) DIV 2;
fact := (fact-1) DIV 2;
repeat
primes[sievePos] := 1;
repeat
dec(fact);
dec(sievePos,sieveprime);
until primes[fact]= 0;
until fact < spIdx;
end;
inc(spIdx);
until false;
end;
{ Not neccessary for this small primes.
procedure EmergencyStop(i:NativeInt);
Begin
Writeln( 'STOP at ',i,'.th prime');
HALT(i);
end;
}

function GetDeltas:NativeUint;
//Converting prime positions into distance
var
i,j,last : NativeInt;
Begin
j :=0;
i := 1;
last :=1;
For i := 1 to High(primes) do
if primes[i] = 0 then
Begin
//IF i-last > 255 {aka delta prim > 512} then EmergencyStop (j);
delta[j] := i-last;
last := i;
inc(j);
end;
GetDeltas := j;
end;
 
procedure OutHeader;
Begin
writeln('Limit':12,'Strong':10,'balanced':12,'weak':10);
end;
 
procedure OutcntWS (const cntWS : tWeakStrong;Lmt:NativeInt);
Begin
with cntWS do
writeln(lmt:12,Strong:10,balanced:12,weak:10);
end;
 
procedure CntWeakStrong10(var Out:tWeakStrong);
// Output a table of values for strang/balanced/weak for 10^n
var
idx,diff,prime,lmt :NativeInt;
begin
OutHeader;
lmt := 10;
fillchar(Out,SizeOf(Out),#0);
idx := 0;
prime:=3;
repeat
dec(prime,2*delta[idx]);
while idx < deltaCnt do
Begin
inc(prime,2*delta[idx]);
IF prime > lmt then
BREAK;
 
diff := delta[idx] - delta[idx+1];
if diff>0 then
inc(Out.strong)
else
if diff< 0 then
inc(Out.weak)
else
inc(Out.balanced);
 
inc(idx);
end;
OutcntWS(Out,Lmt);
lmt := lmt*10;
until Lmt > PrimeLimit;
end;
 
procedure WeakOut(cnt:NativeInt);
var
idx,prime : NativeInt;
begin
Writeln('The first ',cnt,' weak primes');
prime:=3;
idx := 0;
repeat
inc(prime,2*delta[idx]);
if delta[idx] - delta[idx+1]< 0 then
Begin
write(prime,' ');
dec(cnt);
IF cnt <=0 then
BREAK;
end;
inc(idx);
until idx >= deltaCnt;
Writeln;
end;
 
procedure StrongOut(cnt:NativeInt);
var
idx,prime : NativeInt;
begin
Writeln('The first ',cnt,' strong primes');
prime:=3;
idx := 0;
repeat
inc(prime,2*delta[idx]);
if delta[idx] - delta[idx+1]> 0 then
Begin
write(prime,' ');
dec(cnt);
IF cnt <=0 then
BREAK;
end;
inc(idx);
until idx >= deltaCnt;
Writeln;
end;
 
begin
sieveprimes;
deltaCnt := GetDeltas;
 
StrongOut(36);
WeakOut(37);
CntWeakStrong10(CntWs);
end.
Output:
The first 36 strong primes
11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439 
The first 37 weak primes
3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401 
       Limit    Strong    balanced      weak
          10         0           1         2
         100        10           2        12
        1000        73          15        79
       10000       574          65       589
      100000      4543         434      4614
     1000000     37723        2994     37780
    10000000    320991       21837    321750
   100000000   2796946      167032   2797476
  1000000000  24758535     1328401  24760597

real    0m3.011s

Perl[edit]

Translation of: Perl 6
Library: ntheory
use ntheory qw(primes);
 
sub comma {
(my $s = reverse shift) =~ s/(.{3})/$1,/g;
$s =~ s/,(-?)$/$1/;
$s = reverse $s;
}
 
sub below { my($m,@a) = @_; $c = 0; while () { return $c if $a[++$c] > $m } }
 
my @primes = @{ primes(10_000_019) };
 
for $p (1 .. $#primes - 1) {
$x = ($primes[$p - 1] + $primes[$p + 1]) / 2;
if ($x > $primes[$p]) { push @weak, $primes[$p] }
elsif ($x < $primes[$p]) { push @strong, $primes[$p] }
else { push @balanced, $primes[$p] }
}
 
for ([\@strong, 'strong', 36, 1e6, 1e7],
[\@weak, 'weak', 37, 1e6, 1e7],
[\@balanced, 'balanced', 28, 1e6, 1e7]) {
my($pr, $type, $d, $c1, $c2) = @$_;
print "\nFirst $d $type primes:\n", join ' ', map { comma $_ } @$pr[0..$d-1], "\n";
print "Count of $type primes <= @{[comma $c1]}: " . comma below(1e6,@$pr) . "\n";
print "Count of $type primes <= @{[comma $c2]}: " . comma scalar @$pr . "\n";
}
Output:
First 36 strong primes:
11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439
Count of strong primes <=  1,000,000:  37,723
Count of strong primes <= 10,000,000: 320,991

First 37 weak primes:
3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401
Count of weak primes <=  1,000,000:  37,780
Count of weak primes <= 10,000,000: 321,750

First 28 balanced primes:
5 53 157 173 211 257 263 373 563 593 607 653 733 947 977 1,103 1,123 1,187 1,223 1,367 1,511 1,747 1,753 1,907 2,287 2,417 2,677 2,903

Perl 6[edit]

Works with: Rakudo version 2018.11
sub comma { $^i.flip.comb(3).join(',').flip }
 
use Math::Primesieve;
 
my $sieve = Math::Primesieve.new;
 
my @primes = $sieve.primes(10_000_019);
 
my (@weak, @balanced, @strong);
 
for 1 ..^ @primes - 1 -> $p {
given (@primes[$p - 1] + @primes[$p + 1]) / 2 {
when * > @primes[$p] { @weak.push: @primes[$p] }
when * < @primes[$p] { @strong.push: @primes[$p] }
default { @balanced.push: @primes[$p] }
}
}
 
for @strong, 'strong', 36,
@weak, 'weak', 37,
@balanced, 'balanced', 28
-> @pr, $type, $d {
say "\nFirst $d $type primes:\n", @pr[^$d]».&comma;
say "Count of $type primes <= {comma 1e6}: ", comma +@pr[^(@pr.first: * > 1e6,:k)];
say "Count of $type primes <= {comma 1e7}: ", comma +@pr;
}
Output:
First 36 strong primes:
(11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439)
Count of strong primes <=  1,000,000:  37,723
Count of strong primes <= 10,000,000: 320,991

First 37 weak primes:
(3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401)
Count of weak primes <=  1,000,000:  37,780
Count of weak primes <= 10,000,000: 321,750

First 28 balanced primes:
(5 53 157 173 211 257 263 373 563 593 607 653 733 947 977 1,103 1,123 1,187 1,223 1,367 1,511 1,747 1,753 1,907 2,287 2,417 2,677 2,903)
Count of balanced primes <=  1,000,000:  2,994
Count of balanced primes <= 10,000,000: 21,837

Python[edit]

Using the popular numpy library for fast prime generation.

COmputes and shows the requested output then adds similar output for the "balanced" case where prime(p) == [prime(p-1) + prime(p+1)] ÷ 2<code>.

import numpy as np
 
def primesfrom2to(n):
# https://stackoverflow.com/questions/2068372/fastest-way-to-list-all-primes-below-n-in-python/3035188#3035188
""" Input n>=6, Returns a array of primes, 2 <= p < n """
sieve = np.ones(n//3 + (n%6==2), dtype=np.bool)
sieve[0] = False
for i in range(int(n**0.5)//3+1):
if sieve[i]:
k=3*i+1|1
sieve[ ((k*k)//3)  ::2*k] = False
sieve[(k*k+4*k-2*k*(i&1))//3::2*k] = False
return np.r_[2,3,((3*np.nonzero(sieve)[0]+1)|1)]
 
p = primes10m = primesfrom2to(10_000_000)
s = strong10m = [t for s, t, u in zip(p, p[1:], p[2:])
if t > (s + u) / 2]
w = weak10m = [t for s, t, u in zip(p, p[1:], p[2:])
if t < (s + u) / 2]
b = balanced10m = [t for s, t, u in zip(p, p[1:], p[2:])
if t == (s + u) / 2]
 
print('The first 36 strong primes:', s[:36])
print('The count of the strong primes below 1,000,000:',
sum(1 for p in s if p < 1_000_000))
print('The count of the strong primes below 10,000,000:', len(s))
print('\nThe first 37 weak primes:', w[:37])
print('The count of the weak primes below 1,000,000:',
sum(1 for p in w if p < 1_000_000))
print('The count of the weak primes below 10,000,000:', len(w))
print('\n\nThe first 10 balanced primes:', b[:10])
print('The count of balanced primes below 1,000,000:',
sum(1 for p in b if p < 1_000_000))
print('The count of balanced primes below 10,000,000:', len(b))
print('\nTOTAL primes below 1,000,000:',
sum(1 for pr in p if pr < 1_000_000))
print('TOTAL primes below 10,000,000:', len(p))
Output:
The first   36   strong primes: [11, 17, 29, 37, 41, 59, 67, 71, 79, 97, 101, 107, 127, 137, 149, 163, 179, 191, 197, 223, 227, 239, 251, 269, 277, 281, 307, 311, 331, 347, 367, 379, 397, 419, 431, 439]
The   count   of the strong primes below   1,000,000: 37723
The   count   of the strong primes below  10,000,000: 320991

The first   37   weak primes: [3, 7, 13, 19, 23, 31, 43, 47, 61, 73, 83, 89, 103, 109, 113, 131, 139, 151, 167, 181, 193, 199, 229, 233, 241, 271, 283, 293, 313, 317, 337, 349, 353, 359, 383, 389, 401]
The   count   of the weak   primes below   1,000,000: 37780
The   count   of the weak   primes below  10,000,000: 321749


The first   10 balanced primes: [5, 53, 157, 173, 211, 257, 263, 373, 563, 593]
The   count   of balanced   primes below   1,000,000: 2994
The   count   of balanced   primes below  10,000,000: 21837

TOTAL primes below   1,000,000: 78498
TOTAL primes below  10,000,000: 664579

REXX[edit]

/*REXX program lists a sequence  (or a count)  of  ──strong──   or   ──weak──   primes. */
parse arg N kind _ . 1 . okind; upper kind /*obtain optional arguments from the CL*/
if N=='' | N=="," then N= 36 /*Not specified? Then assume default.*/
if kind=='' | kind=="," then kind= 'STRONG' /* " " " " " */
if _\=='' then call ser 'too many arguments specified.'
if kind\=='WEAK' & kind\=='STRONG' then call ser 'invalid 2nd argument: ' okind
if kind =='WEAK' then weak= 1; else weak= 0 /*WEAK is a binary value for function.*/
w = linesize() - 1 /*obtain the usable width of the term. */
tell= (N>0); @.=; N= abs(N) /*N is negative? Then don't display. */
!.=0;  !.1=2;  !.2=3;  !.3=5;  !.4=7;  !.5=11;  !.6=13;  !.7=17;  !.8=19;  !.9=23; #= 8
@.=''; @.2=1; @.3=1; @.5=1; @.7=1; @.11=1; @.13=1; @.17=1; @.19=1; start= # + 1
m= 0; lim= 0 /*# is the number of low primes so far*/
$=; do i=3 for #-2 while lim<=N /* [↓] find primes, and maybe show 'em*/
call strongWeak i-1; $= strip($) /*go see if other part of a KIND prime.*/
end /*i*/ /* [↑] allows faster loop (below). */
/* [↓] N: default lists up to 35 #'s.*/
do j=!.#+2 by 2 while lim<N /*continue on with the next odd prime. */
if j // 3 == 0 then iterate /*is this integer a multiple of three? */
parse var j '' -1 _ /*obtain the last decimal digit of J */
if _ == 5 then iterate /*is this integer a multiple of five? */
if j // 7 == 0 then iterate /* " " " " " " seven? */
if j //11 == 0 then iterate /* " " " " " " eleven?*/
if j //13 == 0 then iterate /* " " " " " " 13 ? */
if j //17 == 0 then iterate /* " " " " " " 17 ? */
if j //19 == 0 then iterate /* " " " " " " 19 ? */
/* [↓] divide by the primes. ___ */
do k=start to # while !.k * !.k<=j /*divide J by other primes ≤ √ J */
if j // !.k ==0 then iterate j /*÷ by prev. prime? ¬prime ___ */
end /*k*/ /* [↑] only divide up to √ J */
#= # + 1 /*bump the count of number of primes. */
 !.#= j; @.j= 1 /*define a prime and its index value.*/
call strongWeak #-1 /*go see if other part of a KIND prime.*/
end /*j*/
/* [↓] display number of primes found.*/
if $\=='' then say $ /*display any residual primes in $ list*/
say
if tell then say commas(m)' ' kind "primes found."
else say commas(m)' ' kind "primes found below or equal to " commas(N).
exit /*stick a fork in it, we're all done. */
/*──────────────────────────────────────────────────────────────────────────────────────*/
add: m= m+1; lim= m; if \tell & y>N then do; lim= y; m= m-1; end; else call app; return 1
app: if tell then if length($ y)>w then do; say $; $= y; end; else $= $ y; return 1
ser: say; say; say '***error***' arg(1); say; say; exit 13 /*tell error message. */
commas: parse arg _; do jc=length(_)-3 to 1 by -3; _=insert(',', _, jc); end; return _
/*──────────────────────────────────────────────────────────────────────────────────────*/
strongWeak: parse arg x; Lp= x - 1; Hp= x + 1; y=!.x; s= (!.Lp + !.Hp) / 2
if weak then if y<s then return add() /*is a weak prime.*/
else return 0 /*not " " " */
else if y>s then return add() /*is an strong prime.*/
return 0 /*not " " " */

This REXX program makes use of   LINESIZE   REXX program (or BIF) which is used to determine the screen width (or linesize) of the terminal (console).   Some REXXes don't have this BIF.

The   LINESIZE.REX   REXX program is included here   ───►   LINESIZE.REX.


output   when using the default input of:     36   strong
11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439

36  STRONG primes found.
output   when using the default input of:     -1000000   strong
37,723  STRONG primes found below or equal to  1,000,000.
output   when using the default input of:     -10000000   strong
320,991  STRONG primes found below or equal to  10,000,000.
output   when using the default input of:     37   weak
3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401

37  WEAK primes found.
output   when using the default input of:     -1000000   weak
37,780  WEAK primes found below or equal to  1,000,000.
output   when using the default input of:     -1000000   weak
321,750  WEAK primes found below or equal to  10,000,000.

zkl[edit]

Using GMP (GNU Multiple Precision Arithmetic Library, probabilistic primes), because it is easy and fast to generate primes.

Extensible prime generator#zkl could be used instead.

var [const] BI=Import("zklBigNum");  // libGMP
const N=1e7;
 
pw,strong,weak := BI(1),List(),List(); // 32,0991 32,1751
ps:=(3).pump(List,'wrap{ pw.nextPrime().toInt() }).copy(); // rolling window
do{
pp,p,pn := ps;
if((z:=(pp.toFloat() + pn)/2)){ // 2,3,5 --> 3.5
if(z>p) weak .append(p);
else if(z<p) strong.append(p);
}
ps.pop(0); ps.append(pw.nextPrime().toInt());
}while(pn<=N);
foreach nm,list,psz in (T(T("strong",strong,36), T("weak",weak,37))){
println("First %d %s primes:\n%s".fmt(psz,nm,list[0,psz].concat(" ")));
println("Count of %s primes <= %,10d: %,8d"
.fmt(nm,1e6,list.reduce('wrap(s,p){ s + (p<=1e6) },0)));
println("Count of %s primes <= %,10d: %,8d\n".fmt(nm,1e7,list.len()));
}
Output:
First 36 strong primes:
11 17 29 37 41 59 67 71 79 97 101 107 127 137 149 163 179 191 197 223 227 239 251 269 277 281 307 311 331 347 367 379 397 419 431 439
Count of strong primes <=  1,000,000:   37,723
Count of strong primes <= 10,000,000:  320,991

First 37 weak primes:
3 7 13 19 23 31 43 47 61 73 83 89 103 109 113 131 139 151 167 181 193 199 229 233 241 271 283 293 313 317 337 349 353 359 383 389 401
Count of weak primes <=  1,000,000:   37,780
Count of weak primes <= 10,000,000:  321,750