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
-
- The OEIS article A051634: strong primes.
- The OEIS article A051635: weak primes.
11l
F primes_upto(limit)
V is_prime = [0B] * 2 [+] [1B] * (limit - 1)
L(n) 0 .< Int(limit ^ 0.5 + 1.5)
I is_prime[n]
L(i) (n * n .< limit + 1).step(n)
is_prime[i] = 0B
R enumerate(is_prime).filter((i, prime) -> prime).map((i, prime) -> i)
V p = primes_upto(10'000'000)
[Int] s, w, b
L(i) 1 .< p.len - 1
I p[i] > (p[i - 1] + p[i + 1]) * 0.5
s [+]= p[i]
E I p[i] < (p[i - 1] + p[i + 1]) * 0.5
w [+]= p[i]
E
b [+]= p[i]
print(‘The first 36 strong primes: ’s[0.<36])
print(‘The count of the strong primes below 1,000,000: ’sum(s.filter(p -> p < 1'000'000).map(p -> 1)))
print(‘The count of the strong primes below 10,000,000: ’s.len)
print("\nThe first 37 weak primes: "w[0.<37])
print(‘The count of the weak primes below 1,000,000: ’sum(w.filter(p -> p < 1'000'000).map(p -> 1)))
print(‘The count of the weak primes below 10,000,000: ’w.len)
print("\n\nThe first 10 balanced primes: "b[0.<10])
print(‘The count of balanced primes below 1,000,000: ’sum(b.filter(p -> p < 1'000'000).map(p -> 1)))
print(‘The count of balanced primes below 10,000,000: ’b.len)
print("\nTOTAL primes below 1,000,000: "sum(p.filter(pr -> pr < 1'000'000).map(pr -> 1)))
print(‘TOTAL primes below 10,000,000: ’p.len)- 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
ALGOL 68
# 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
AWK
# syntax: GAWK -f STRONG_AND_WEAK_PRIMES.AWK
BEGIN {
for (i=1; i<1E7; i++) {
if (is_prime(i)) {
arr[++n] = i
}
}
# strong:
stop1 = 36 ; stop2 = 1E6 ; stop3 = 1E7
count1 = count2 = count3 = 0
printf("The first %d strong primes:",stop1)
for (i=2; count1<stop1; i++) {
if (arr[i] > (arr[i-1] + arr[i+1]) / 2) {
count1++
printf(" %d",arr[i])
}
}
printf("\n")
for (i=2; i<stop3; i++) {
if (arr[i] > (arr[i-1] + arr[i+1]) / 2) {
count3++
if (arr[i] < stop2) {
count2++
}
}
}
printf("Number below %d: %d\n",stop2,count2)
printf("Number below %d: %d\n",stop3,count3)
# weak:
stop1 = 37 ; stop2 = 1E6 ; stop3 = 1E7
count1 = count2 = count3 = 0
printf("The first %d weak primes:",stop1)
for (i=2; count1<stop1; i++) {
if (arr[i] < (arr[i-1] + arr[i+1]) / 2) {
count1++
printf(" %d",arr[i])
}
}
printf("\n")
for (i=2; i<stop3; i++) {
if (arr[i] < (arr[i-1] + arr[i+1]) / 2) {
count3++
if (arr[i] < stop2) {
count2++
}
}
}
printf("Number below %d: %d\n",stop2,count2)
printf("Number below %d: %d\n",stop3,count3)
exit(0)
}
function is_prime(n, d) {
d = 5
if (n < 2) { return(0) }
if (n % 2 == 0) { return(n == 2) }
if (n % 3 == 0) { return(n == 3) }
while (d*d <= n) {
if (n % d == 0) { return(0) }
d += 2
if (n % d == 0) { return(0) }
d += 4
}
return(1)
}
- 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 Number below 1000000: 37723 Number below 10000000: 320992 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 Number below 1000000: 37781 Number below 10000000: 321750
C
#include <stdbool.h>
#include <stdint.h>
#include <stdio.h>
#include <stdlib.h>
const int PRIMES[] = {
2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293,
307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523,
541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769,
773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997,
1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217,
1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451,
1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663,
1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907,
1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141,
2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383,
2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659,
2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861,
2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163
};
#define PRIME_LENGTH (sizeof(PRIMES) / sizeof(int))
bool isPrime(int n) {
int i;
if (n < 2) {
return false;
}
for (i = 0; i < PRIME_LENGTH; ++i) {
if (n == PRIMES[i]) {
return true;
}
if (n % PRIMES[i] == 0) {
return false;
}
if (n < PRIMES[i] * PRIMES[i]) {
break;
}
}
return true;
}
int main() {
const int MAX_LENGTH = 700000;
int i, n, c1, c2;
int *primePtr = calloc(MAX_LENGTH, sizeof(int));
if (primePtr == 0) {
return EXIT_FAILURE;
}
for (i = 0; i < PRIME_LENGTH; i++) {
primePtr[i] = PRIMES[i];
}
i--;
for (n = PRIMES[i] + 4; n < 10000100;) {
if (isPrime(n)) {
primePtr[i++] = n;
}
n += 2;
if (isPrime(n)) {
primePtr[i++] = n;
}
n += 4;
if (i >= MAX_LENGTH) {
printf("Allocate more memory.");
return EXIT_FAILURE;
}
}
/////////////////////////////////////////////////////////////
printf("First 36 strong primes:");
c1 = 0;
c2 = 0;
for (n = 0, i = 1; i < MAX_LENGTH - 1; i++) {
if (2 * primePtr[i] > primePtr[i - 1] + primePtr[i + 1]) {
if (n < 36) {
printf(" %d", primePtr[i]);
n++;
}
if (primePtr[i] < 1000000) {
c1++;
c2++;
} else if (primePtr[i] < 10000000) {
c2++;
} else break;
}
}
printf("\nThere are %d strong primes below 1,000,000", c1);
printf("\nThere are %d strong primes below 10,000,000\n\n", c2);
/////////////////////////////////////////////////////////////
printf("First 37 weak primes:");
c1 = 0;
c2 = 0;
for (n = 0, i = 1; i < MAX_LENGTH - 1; i++) {
if (2 * primePtr[i] < primePtr[i - 1] + primePtr[i + 1]) {
if (n < 37) {
printf(" %d", primePtr[i]);
n++;
}
if (primePtr[i] < 1000000) {
c1++;
c2++;
} else if (primePtr[i] < 10000000) {
c2++;
} else break;
}
}
printf("\nThere are %d weak primes below 1,000,000", c1);
printf("\nThere are %d weak primes below 10,000,000\n\n", c2);
free(primePtr);
return EXIT_SUCCESS;
}
- 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 There are 37722 strong primes below 1,000,000 There are 320990 strong primes below 10,000,000 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 There are 37780 weak primes below 1,000,000 There are 321750 weak primes below 10,000,000
C#
using static System.Console;
using static System.Linq.Enumerable;
using System;
public static class StrongAndWeakPrimes
{
public static void Main() {
var primes = PrimeGenerator(10_000_100).ToList();
var strongPrimes = from i in Range(1, primes.Count - 2) where primes[i] > (primes[i-1] + primes[i+1]) / 2 select primes[i];
var weakPrimes = from i in Range(1, primes.Count - 2) where primes[i] < (primes[i-1] + primes[i+1]) / 2.0 select primes[i];
WriteLine($"First 36 strong primes: {string.Join(", ", strongPrimes.Take(36))}");
WriteLine($"There are {strongPrimes.TakeWhile(p => p < 1_000_000).Count():N0} strong primes below {1_000_000:N0}");
WriteLine($"There are {strongPrimes.TakeWhile(p => p < 10_000_000).Count():N0} strong primes below {10_000_000:N0}");
WriteLine($"First 37 weak primes: {string.Join(", ", weakPrimes.Take(37))}");
WriteLine($"There are {weakPrimes.TakeWhile(p => p < 1_000_000).Count():N0} weak primes below {1_000_000:N0}");
WriteLine($"There are {weakPrimes.TakeWhile(p => p < 10_000_000).Count():N0} weak primes below {1_000_000:N0}");
}
}
- 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 There are 37,723 strong primes below 1,000,000 There are 320,991 strong primes below 10,000,000 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 There are 37,780 weak primes below 1,000,000 There are 321,750 weak primes below 1,000,000
C++
#include <algorithm>
#include <iostream>
#include <iterator>
#include <locale>
#include <vector>
#include "prime_sieve.hpp"
const int limit1 = 1000000;
const int limit2 = 10000000;
class prime_info {
public:
explicit prime_info(int max) : max_print(max) {}
void add_prime(int prime);
void print(std::ostream& os, const char* name) const;
private:
int max_print;
int count1 = 0;
int count2 = 0;
std::vector<int> primes;
};
void prime_info::add_prime(int prime) {
++count2;
if (prime < limit1)
++count1;
if (count2 <= max_print)
primes.push_back(prime);
}
void prime_info::print(std::ostream& os, const char* name) const {
os << "First " << max_print << " " << name << " primes: ";
std::copy(primes.begin(), primes.end(), std::ostream_iterator<int>(os, " "));
os << '\n';
os << "Number of " << name << " primes below " << limit1 << ": " << count1 << '\n';
os << "Number of " << name << " primes below " << limit2 << ": " << count2 << '\n';
}
int main() {
prime_sieve sieve(limit2 + 100);
// write numbers with groups of digits separated according to the system default locale
std::cout.imbue(std::locale(""));
// count and print strong/weak prime numbers
prime_info strong_primes(36);
prime_info weak_primes(37);
int p1 = 2, p2 = 3;
for (int p3 = 5; p2 < limit2; ++p3) {
if (!sieve.is_prime(p3))
continue;
int diff = p1 + p3 - 2 * p2;
if (diff < 0)
strong_primes.add_prime(p2);
else if (diff > 0)
weak_primes.add_prime(p2);
p1 = p2;
p2 = p3;
}
strong_primes.print(std::cout, "strong");
weak_primes.print(std::cout, "weak");
return 0;
}
Contents of prime_sieve.hpp:
#ifndef PRIME_SIEVE_HPP
#define PRIME_SIEVE_HPP
#include <algorithm>
#include <vector>
/**
* A simple implementation of the Sieve of Eratosthenes.
* See https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes.
*/
class prime_sieve {
public:
explicit prime_sieve(size_t);
bool is_prime(size_t) const;
private:
std::vector<bool> is_prime_;
};
/**
* Constructs a sieve with the given limit.
*
* @param limit the maximum integer that can be tested for primality
*/
inline prime_sieve::prime_sieve(size_t limit) {
limit = std::max(size_t(3), limit);
is_prime_.resize(limit/2, true);
for (size_t p = 3; p * p <= limit; p += 2) {
if (is_prime_[p/2 - 1]) {
size_t inc = 2 * p;
for (size_t q = p * p; q <= limit; q += inc)
is_prime_[q/2 - 1] = false;
}
}
}
/**
* Returns true if the given integer is a prime number. The integer
* must be less than or equal to the limit passed to the constructor.
*
* @param n an integer less than or equal to the limit passed to the
* constructor
* @return true if the integer is prime
*/
inline bool prime_sieve::is_prime(size_t n) const {
if (n == 2)
return true;
if (n < 2 || n % 2 == 0)
return false;
return is_prime_.at(n/2 - 1);
}
#endif
- 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 Number of strong primes below 1,000,000: 37,723 Number of 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 Number of weak primes below 1,000,000: 37,780 Number of weak primes below 10,000,000: 321,750
D
import std.algorithm;
import std.array;
import std.range;
import std.stdio;
immutable PRIMES = [
2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97,
101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293,
307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503, 509, 521, 523,
541, 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, 739, 743, 751, 757, 761, 769,
773, 787, 797, 809, 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, 947, 953, 967, 971, 977, 983, 991, 997,
1009, 1013, 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217,
1223, 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451,
1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, 1663,
1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, 1901, 1907,
1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, 2131, 2137, 2141,
2143, 2153, 2161, 2179, 2203, 2207, 2213, 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, 2371, 2377, 2381, 2383,
2389, 2393, 2399, 2411, 2417, 2423, 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, 2621, 2633, 2647, 2657, 2659,
2663, 2671, 2677, 2683, 2687, 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, 2833, 2837, 2843, 2851, 2857, 2861,
2879, 2887, 2897, 2903, 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, 3083, 3089, 3109, 3119, 3121, 3137, 3163,
3167, 3169, 3181, 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391,
3407, 3413, 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637,
3643, 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907,
3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, 4153,
4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, 4421, 4423,
4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, 4663, 4673, 4679,
4691, 4703, 4721, 4723, 4729, 4733, 4751, 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, 4943, 4951, 4957, 4967,
4969, 4973, 4987, 4993, 4999, 5003, 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, 5189, 5197, 5209, 5227, 5231,
5233, 5237, 5261, 5273, 5279, 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, 5449, 5471, 5477, 5479, 5483, 5501,
5503, 5507, 5519, 5521, 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, 5701, 5711, 5717, 5737, 5741, 5743, 5749,
5779, 5783, 5791, 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043,
6047, 6053, 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299,
6301, 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571,
6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, 6841,
6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, 7109, 7121,
7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, 7417, 7433, 7451,
7457, 7459, 7477, 7481, 7487, 7489, 7499, 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, 7649, 7669, 7673, 7681,
7687, 7691, 7699, 7703, 7717, 7723, 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, 7927, 7933, 7937, 7949, 7951,
7963, 7993, 8009, 8011, 8017, 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, 8221, 8231, 8233, 8237, 8243, 8263,
8269, 8273, 8287, 8291, 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, 8513, 8521, 8527, 8537, 8539, 8543, 8563,
8573, 8581, 8597, 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819,
8821, 8831, 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103,
9109, 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377,
9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, 9643,
9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, 9901, 9907,
9923, 9929, 9931, 9941, 9949, 9967, 9973,
];
bool isPrime(int n) {
if (n < 2) {
return false;
}
foreach (prime; PRIMES) {
if (n == prime) {
return true;
}
if (n % prime == 0) {
return false;
}
if (n < prime * prime) {
if (n > PRIMES[$-1] * PRIMES[$-1]) {
assert(false, "Out of pre-computed primes.");
}
break;
}
}
return true;
}
void main() {
auto primeList = iota(2, 10_000_100).filter!isPrime.array;
int[] strongPrimes, weakPrimes;
foreach (i,p; primeList) {
if (i > 0 && i < primeList.length - 1) {
if (p > 0.5 * (primeList[i - 1] + primeList[i + 1])) {
strongPrimes ~= p;
} else if (p < 0.5 * (primeList[i - 1] + primeList[i + 1])) {
weakPrimes ~= p;
}
}
}
writeln("First 36 strong primes: ", strongPrimes[0..36]);
writefln("There are %d strong primes below 1,000,000", strongPrimes.filter!"a<1_000_000".count);
writefln("There are %d strong primes below 10,000,000", strongPrimes.filter!"a<10_000_000".count);
writeln("First 37 weak primes: ", weakPrimes[0..37]);
writefln("There are %d weak primes below 1,000,000", weakPrimes.filter!"a<1_000_000".count);
writefln("There are %d weak primes below 10,000,000", weakPrimes.filter!"a<10_000_000".count);
}
- 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] There are 37723 strong primes below 1,000,000 There are 320991 strong primes below 10,000,000 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] There are 37780 weak primes below 1,000,000 There are 321750 weak primes below 10,000,000
Delphi
procedure StrongWeakPrimes(Memo: TMemo);
{Display Strong/Weak prime information}
var I,P: integer;
var Sieve: TPrimeSieve;
var S: string;
var Cnt,Cnt1,Cnt2: integer;
type TPrimeTypes = (ptStrong,ptWeak,ptBalanced);
function GetTypeStr(PrimeType: TPrimeTypes): string;
{Get string describing PrimeType}
begin
case PrimeType of
ptStrong: Result:='Strong';
ptWeak: Result:='Weak';
ptBalanced: Result:='Balanced';
end;
end;
function GetPrimeType(N: integer): TPrimeTypes;
{Return flag indicating type of prime Primes[N] is}
{Strong = Primes(N) > [Primes(N-1) + Primes(N+1)] / 2}
{Weak = Primes(N) < [Primes(N-1) + Primes(N+1)] / 2}
{Balanced = Primes(N) = [Primes(N-1) + Primes(N+1)] / 2}
var P,P1: double;
begin
P:=Sieve.Primes[N];
P1:=(Sieve.Primes[N-1] + Sieve.Primes[N+1]) / 2;
if P>P1 then Result:=ptStrong
else if P<P1 then Result:=ptWeak
else Result:=ptBalanced;
end;
procedure GetPrimeCounts(PT: TPrimeTypes; var Cnt1,Cnt2: integer);
{Get number of primes of type "PT" below 1 million and 10 million}
var I: integer;
begin
Cnt1:=0; Cnt2:=0;
for I:=1 to 1000000-1 do
begin
if GetPrimeType(I)=PT then
begin
if Sieve.Primes[I]>10000000 then break;
Inc(Cnt2);
if Sieve.Primes[I]<1000000 then Inc(Cnt1);
end;
end;
end;
function GetPrimeList(PT: TPrimeTypes; Limit: integer): string;
{Get a list of primes of type PT up to Limit}
var I,Cnt: integer;
begin
Result:='';
Cnt:=0;
for I:=1 to Sieve.PrimeCount-1 do
if GetPrimeType(I)=PT then
begin
Inc(Cnt);
P:=Sieve.Primes[I];
Result:=Result+Format('%5d',[P]);
if Cnt>=Limit then break;
if (Cnt mod 10)=0 then Result:=Result+CRLF;
end;
end;
procedure ShowPrimeTypeData(PT: TPrimeTypes; Limit: Integer);
{Display information about specified PrimeType, listing items up to Limit}
var S,TS: string;
begin
S:=GetPrimeList(PT,Limit);
TS:=GetTypeStr(PT);
Memo.Lines.Add(Format('First %d %s primes are:',[Limit,TS]));
Memo.Lines.Add(S);
GetPrimeCounts(PT,Cnt1,Cnt2);
Memo.Lines.Add(Format('Number %s primes <1,000,000: %8.0n', [TS,Cnt1+0.0]));
Memo.Lines.Add(Format('Number %s primes <10,000,000: %8.0n', [TS,Cnt2+0.0]));
Memo.Lines.Add('');
end;
begin
Sieve:=TPrimeSieve.Create;
try
Sieve.Intialize(200000000);
Memo.Lines.Add('Primes in Sieve : '+IntToStr(Sieve.PrimeCount));
ShowPrimeTypeData(ptStrong,36);
ShowPrimeTypeData(ptWeak,37);
ShowPrimeTypeData(ptBalanced,28);
finally Sieve.Free; end;
end;
- Output:
Primes in Sieve : 11078937
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
Number Strong primes <1,000,000: 37,723
Number Strong primes <10,000,000: 320,991
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
Number Weak primes <1,000,000: 37,780
Number Weak primes <10,000,000: 321,750
First 28 Balanced primes are:
5 53 157 173 211 257 263 373 563 593
607 653 733 947 977 1103 1123 1187 1223 1367
1511 1747 1753 1907 2287 2417 2677 2903
Number Balanced primes <1,000,000: 2,994
Number Balanced primes <10,000,000: 21,837
Elapsed Time: 2.947 Sec.
EasyLang
fastfunc isprim num .
i = 3
while i <= sqrt num
if num mod i = 0
return 0
.
i += 2
.
return 1
.
func nextprim n .
repeat
n += 2
until isprim n = 1
.
return n
.
proc strwkprimes ncnt sgn . .
write "First " & ncnt & ": "
pr2 = 2
pr3 = 3
repeat
pr1 = pr2
pr2 = pr3
until pr2 >= 10000000
pr3 = nextprim pr3
if pr1 < 1000000 and pr2 > 1000000
print ""
print "Count lower 10e6: " & cnt
.
if sgn * pr2 > sgn * (pr1 + pr3) / 2
cnt += 1
if cnt <= ncnt
write pr2 & " "
.
.
.
print "Count lower 10e7: " & cnt
print ""
.
print "Strong primes:"
strwkprimes 36 1
print "Weak primes:"
strwkprimes 37 -1
- Output:
Strong primes: First 36: 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 lower 10e6: 37723 Count lower 10e7: 320991 Weak primes: First 37: 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 lower 10e6: 37780 Count lower 10e7: 321750
Factor
USING: formatting grouping kernel math math.primes sequences
tools.memory.private ;
IN: rosetta-code.strong-primes
: fn ( p-1 p p+1 -- p sum ) rot + 2 / ;
: strong? ( p-1 p p+1 -- ? ) fn > ;
: weak? ( p-1 p p+1 -- ? ) fn < ;
: swprimes ( seq quot -- seq )
[ 3 <clumps> ] dip [ first3 ] prepose filter [ second ] map
; inline
: stats ( seq n -- firstn count1 count2 )
[ head ] [ drop [ 1e6 < ] filter length ] [ drop length ]
2tri [ commas ] bi@ ;
10,000,019 primes-upto [ strong? ] over [ weak? ]
[ swprimes ] 2bi@ [ 36 ] [ 37 ] bi* [ stats ] 2bi@
"First 36 strong primes:\n%[%d, %]
%s strong primes below 1,000,000
%s strong primes below 10,000,000\n
First 37 weak primes:\n%[%d, %]
%s weak primes below 1,000,000
%s weak primes below 10,000,000\n" printf
- 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 }
37,723 strong primes below 1,000,000
320,991 strong primes below 10,000,000
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 }
37,780 weak primes below 1,000,000
321,750 weak primes below 10,000,000
FreeBASIC
#include "isprime.bas"
function nextprime( n as uinteger ) as uinteger
'finds the next prime after n, excluding n if it happens to be prime itself
if n = 0 then return 2
if n < 3 then return n + 1
dim as integer q = n + 2
while not isprime(q)
q+=2
wend
return q
end function
function lastprime( n as uinteger ) as uinteger
'finds the last prime before n, excluding n if it happens to be prime itself
if n = 2 then return 0 'zero isn't prime, but it is a good sentinel value :)
if n = 3 then return 2
dim as integer q = n - 2
while not isprime(q)
q-=2
wend
return q
end function
function isstrong( p as integer ) as boolean
if nextprime(p) + lastprime(p) >= 2*p then return false else return true
end function
function isweak( p as integer ) as boolean
if nextprime(p) + lastprime(p) <= 2*p then return false else return true
end function
print "The first 36 strong primes are: "
dim as uinteger c, p=3
while p < 10000000
if isprime(p) andalso isstrong(p) then
c += 1
if c <= 36 then print p;" ";
if c=37 then print
end if
if p = 1000001 then print "There are ";c;" strong primes below one million"
p+=2
wend
print "There are ";c;" strong primes below ten million"
print
print "The first 37 weak primes are: "
p=3 : c=0
while p < 10000000
if isprime(p) andalso isweak(p) then
c += 1
if c <= 37 then print p;" ";
if c=38 then print
end if
if p = 1000001 then print "There are ";c;" weak primes below one million"
p+=2
wend
print "There are ";c;" weak primes below ten million"
print
- 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
Frink
strongPrimes[end=undef] := select[primes[3,end], {|p| p > (previousPrime[p] + nextPrime[p])/2 }]
weakPrimes[end=undef] := select[primes[3,end], {|p| p < (previousPrime[p] + nextPrime[p])/2 }]
println["First 36 strong primes: " + first[strongPrimes[], 36]]
println["Strong primes below 1,000,000: " + length[strongPrimes[1_000_000]]]
println["Strong primes below 10,000,000: " + length[strongPrimes[10_000_000]]]
println["First 37 weak primes: " + first[weakPrimes[], 37]]
println["Weak primes below 1,000,000: " + length[weakPrimes[1_000_000]]]
println["Weak primes below 10,000,000: " + length[weakPrimes[10_000_000]]]- 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: 37723 Strong primes below 10,000,000: 320991 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: 37780 Weak primes below 10,000,000: 321750
Go
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
Haskell
Uses primes library: http://hackage.haskell.org/package/primes-0.2.1.0/docs/Data-Numbers-Primes.html
import Text.Printf (printf)
import Data.Numbers.Primes (primes)
xPrimes :: (Real a, Fractional b) => (b -> b -> Bool) -> [a] -> [a]
xPrimes op ps@(p1:p2:p3:xs)
| realToFrac p2 `op` (realToFrac (p1 + p3) / 2) = p2 : xPrimes op (tail ps)
| otherwise = xPrimes op (tail ps)
main :: IO ()
main = do
printf "First 36 strong primes: %s\n" . show . take 36 $ strongPrimes
printf "Strong primes below 1,000,000: %d\n" . length . takeWhile (<1000000) $ strongPrimes
printf "Strong primes below 10,000,000: %d\n\n" . length . takeWhile (<10000000) $ strongPrimes
printf "First 37 weak primes: %s\n" . show . take 37 $ weakPrimes
printf "Weak primes below 1,000,000: %d\n" . length . takeWhile (<1000000) $ weakPrimes
printf "Weak primes below 10,000,000: %d\n\n" . length . takeWhile (<10000000) $ weakPrimes
where strongPrimes = xPrimes (>) primes
weakPrimes = xPrimes (<) primes
- 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: 37723 Strong primes below 10,000,000: 320991 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: 37780 Weak primes below 10,000,000: 321750
J
Filter =: (#~`)(`:6)
average =: +/ % #
NB. vector of primes from 2 to 10000019
PRIMES=:i.@>:&.(p:inv) 10000000
strongQ =: 1&{ > [: average {. , {:
STRONG_PRIMES=: (0, 0,~ 3&(strongQ\))Filter PRIMES
NB. first 36 strong primes
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
NB. tally of strong primes less than one and ten million
+/ STRONG_PRIMES </ 1e6 * 1 10
37723 320991
weakQ =: 1&{ < [: average {. , {:
weaklings =: (0, 0,~ 3&(weakQ\))Filter PRIMES
NB. first 37 weak primes
37 {. weaklings
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
NB. tally of weak primes less than one and ten million
+/ weaklings </ 1e6 * 1 10
37780 321750
Java
public class StrongAndWeakPrimes {
private static int MAX = 10_000_000 + 1000;
private static boolean[] primes = new boolean[MAX];
public static void main(String[] args) {
sieve();
System.out.println("First 36 strong primes:");
displayStrongPrimes(36);
for ( int n : new int[] {1_000_000, 10_000_000}) {
System.out.printf("Number of strong primes below %,d = %,d%n", n, strongPrimesBelow(n));
}
System.out.println("First 37 weak primes:");
displayWeakPrimes(37);
for ( int n : new int[] {1_000_000, 10_000_000}) {
System.out.printf("Number of weak primes below %,d = %,d%n", n, weakPrimesBelow(n));
}
}
private static int weakPrimesBelow(int maxPrime) {
int priorPrime = 2;
int currentPrime = 3;
int count = 0;
while ( currentPrime < maxPrime ) {
int nextPrime = getNextPrime(currentPrime);
if ( currentPrime * 2 < priorPrime + nextPrime ) {
count++;
}
priorPrime = currentPrime;
currentPrime = nextPrime;
}
return count;
}
private static void displayWeakPrimes(int maxCount) {
int priorPrime = 2;
int currentPrime = 3;
int count = 0;
while ( count < maxCount ) {
int nextPrime = getNextPrime(currentPrime);
if ( currentPrime * 2 < priorPrime + nextPrime) {
count++;
System.out.printf("%d ", currentPrime);
}
priorPrime = currentPrime;
currentPrime = nextPrime;
}
System.out.println();
}
private static int getNextPrime(int currentPrime) {
int nextPrime = currentPrime + 2;
while ( ! primes[nextPrime] ) {
nextPrime += 2;
}
return nextPrime;
}
private static int strongPrimesBelow(int maxPrime) {
int priorPrime = 2;
int currentPrime = 3;
int count = 0;
while ( currentPrime < maxPrime ) {
int nextPrime = getNextPrime(currentPrime);
if ( currentPrime * 2 > priorPrime + nextPrime ) {
count++;
}
priorPrime = currentPrime;
currentPrime = nextPrime;
}
return count;
}
private static void displayStrongPrimes(int maxCount) {
int priorPrime = 2;
int currentPrime = 3;
int count = 0;
while ( count < maxCount ) {
int nextPrime = getNextPrime(currentPrime);
if ( currentPrime * 2 > priorPrime + nextPrime) {
count++;
System.out.printf("%d ", currentPrime);
}
priorPrime = currentPrime;
currentPrime = nextPrime;
}
System.out.println();
}
private static final void sieve() {
// primes
for ( int i = 2 ; i < MAX ; i++ ) {
primes[i] = true;
}
for ( int i = 2 ; i < MAX ; i++ ) {
if ( primes[i] ) {
for ( int j = 2*i ; j < MAX ; j += i ) {
primes[j] = false;
}
}
}
}
}
- 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 Number of strong primes below 1,000,000 = 37,723 Number of 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 Number of weak primes below 1,000,000 = 37,780 Number of weak primes below 10,000,000 = 321,750
jq
Also works with gojq, the Go implementation of jq
The following assumes that `primes` generates a stream of primes less than or equal to `.` as defined, for example, at Sieve of Eratosthenes]].
def count(s): reduce s as $_ (0; .+1);
# Emit {strong, weak} primes up to and including $n
def strong_weak_primes:
. as $n
| primes as $primes
| ("\nCheck: last prime generated: \($primes[-1])" | debug) as $debug
| reduce range(1; $primes|length-1) as $p ({};
(($primes[$p-1] + $primes[$p+1]) / 2) as $x
| if $primes[$p] > $x
then .strong += [$primes[$p]]
elif $primes[$p] < $x
then .weak += [$primes[$p]]
else .
end );
(1e7 + 19)
| strong_weak_primes as {$strong, $weak}
| "The first 36 strong primes are:",
$strong[:36],
"\nThe count of the strong primes below 1e6: \(count($strong[]|select(. < 1e6 )))",
"\nThe count of the strong primes below 1e7: \(count($strong[]|select(. < 1e7 )))",
"\nThe first 37 weak primes are:",
$weak[:37],
"\nThe count of the weak primes below 1e6: \(count($weak[]|select(. < 1e6 )))",
"\nThe count of the weak primes below 1e7: \(count($weak[]|select(. < 1e7 )))"- 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 count of the strong primes below 1e6: 37723 The count of the strong primes below 1e7: 320991 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 count of the weak primes below 1e6: 37780 The count of the weak primes below 1e7: 321750
Julia
using Primes, Formatting
function parseprimelist()
primelist = primes(2, 10000019)
strongs = Vector{Int64}()
weaks = Vector{Int64}()
balanceds = Vector{Int64}()
for (n, p) in enumerate(primelist)
if n == 1 || n == length(primelist)
continue
end
x = (primelist[n - 1] + primelist[n + 1]) / 2
if x > p
push!(weaks, p)
elseif x < p
push!(strongs, p)
else
push!(balanceds, p)
end
end
println("The first 36 strong primes are: ", strongs[1:36])
println("There are ", format(sum(map(x -> x < 1000000, strongs)), commas=true), " stromg primes less than 1 million.")
println("There are ", format(length(strongs), commas=true), " strong primes less than 10 million.")
println("The first 37 weak primes are: ", weaks[1:37])
println("There are ", format(sum(map(x -> x < 1000000, weaks)), commas=true), " weak primes less than 1 million.")
println("There are ", format(length(weaks), commas=true), " weak primes less than 10 million.")
println("The first 28 balanced primes are: ", balanceds[1:28])
println("There are ", format(sum(map(x -> x < 1000000, balanceds)), commas=true), " balanced primes less than 1 million.")
println("There are ", format(length(balanceds), commas=true), " balanced primes less than 10 million.")
end
parseprimelist()
- 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 37,723 stromg primes less than 1 million. There are 320,991 strong primes less than 10 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 37,780 weak primes less than 1 million. There are 321,750 weak primes less than 10 million. The first 28 balanced primes are: [5, 53, 157, 173, 211, 257, 263, 373, 563, 593, 607, 653, 733, 947, 977, 1103, 1123, 1187, 1223, 1367, 1511, 1747, 1753, 1907, 2287, 2417, 2677, 2903] There are 2,994 balanced primes less than 1 million. There are 21,837 balanced primes less than 10 million.
Kotlin
private const val MAX = 10000000 + 1000
private val primes = BooleanArray(MAX)
fun main() {
sieve()
println("First 36 strong primes:")
displayStrongPrimes(36)
for (n in intArrayOf(1000000, 10000000)) {
System.out.printf("Number of strong primes below %,d = %,d%n", n, strongPrimesBelow(n))
}
println("First 37 weak primes:")
displayWeakPrimes(37)
for (n in intArrayOf(1000000, 10000000)) {
System.out.printf("Number of weak primes below %,d = %,d%n", n, weakPrimesBelow(n))
}
}
private fun weakPrimesBelow(maxPrime: Int): Int {
var priorPrime = 2
var currentPrime = 3
var count = 0
while (currentPrime < maxPrime) {
val nextPrime = getNextPrime(currentPrime)
if (currentPrime * 2 < priorPrime + nextPrime) {
count++
}
priorPrime = currentPrime
currentPrime = nextPrime
}
return count
}
private fun displayWeakPrimes(maxCount: Int) {
var priorPrime = 2
var currentPrime = 3
var count = 0
while (count < maxCount) {
val nextPrime = getNextPrime(currentPrime)
if (currentPrime * 2 < priorPrime + nextPrime) {
count++
print("$currentPrime ")
}
priorPrime = currentPrime
currentPrime = nextPrime
}
println()
}
private fun getNextPrime(currentPrime: Int): Int {
var nextPrime = currentPrime + 2
while (!primes[nextPrime]) {
nextPrime += 2
}
return nextPrime
}
private fun strongPrimesBelow(maxPrime: Int): Int {
var priorPrime = 2
var currentPrime = 3
var count = 0
while (currentPrime < maxPrime) {
val nextPrime = getNextPrime(currentPrime)
if (currentPrime * 2 > priorPrime + nextPrime) {
count++
}
priorPrime = currentPrime
currentPrime = nextPrime
}
return count
}
private fun displayStrongPrimes(maxCount: Int) {
var priorPrime = 2
var currentPrime = 3
var count = 0
while (count < maxCount) {
val nextPrime = getNextPrime(currentPrime)
if (currentPrime * 2 > priorPrime + nextPrime) {
count++
print("$currentPrime ")
}
priorPrime = currentPrime
currentPrime = nextPrime
}
println()
}
private fun sieve() { // primes
for (i in 2 until MAX) {
primes[i] = true
}
for (i in 2 until MAX) {
if (primes[i]) {
var j = 2 * i
while (j < MAX) {
primes[j] = false
j += i
}
}
}
}
- 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 Number of strong primes below 1,000,000 = 37,723 Number of 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 Number of weak primes below 1,000,000 = 37,780 Number of weak primes below 10,000,000 = 321,750
Ksh
#!/bin/ksh
# Strong and weak primes
# # 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. ???
# # A strong prime is when prime[p] > (prime[p-1] + prime[p+1]) ÷ 2
# # A weak prime is when prime[p] < (prime[p-1] + prime[p+1]) ÷ 2
# # Balanced prime is when prime[p] = (prime[p-1] + prime[p+1]) ÷ 2
# # Variables:
#
integer NUM_STRONG=36 NUM_WEAK=37 GOAL1=1000000 MAX_INT=10000000
# # Functions:
#
# # Function _isprime(n) return 1 for prime, 0 for not prime
#
function _isprime {
typeset _n ; integer _n=$1
typeset _i ; integer _i
(( _n < 2 )) && return 0
for (( _i=2 ; _i*_i<=_n ; _i++ )); do
(( ! ( _n % _i ) )) && return 0
done
return 1
}
# # Function _strength(prime[n], prime[n-1], prime[n+1]) return 1 for strong
#
function _strength {
typeset _pri ; integer _pri=$1 # PRIme number under consideration
typeset _pre ; integer _pre=$2 # PREvious prime number
typeset _nex ; integer _nex=$3 # NEXt prime number
typeset _result ; typeset -F1 _result
(( _result = (_pre + _nex) / 2.0 ))
(( _pri > _result )) && echo STRONG && return 0
(( _pri < _result )) && echo WEAK && return 1
echo BALANCED && return 99
}
#####
# main #
######
integer spcnt=0 wpcnt=0 bpcnt=0 sflg=0 wflg=0 i j k goal1_strong goal1_weak
typeset -C prime # prime[].val prime[].typ
typeset -a prime.val
typeset -a prime.typ
prime.typ[0]='NA' ; prime.typ[1]='NA'
for (( i=2; i<MAX_INT; i++ )); do
_isprime ${i} ; (( ! $? )) && continue
prime.val+=( ${i} )
(( ${#prime.val[*]} <= 2 )) && continue
(( j = ${#prime.val[*]} - 2 )) ; (( k = j - 1 ))
prime.typ+=( $(_strength ${prime.val[${j}]} ${prime.val[k]} ${prime.val[-1]}) )
case $? in
0) (( spcnt++ ))
(( spcnt <= NUM_STRONG )) && strbuff+="${prime.val[j]}, "
(( i >= GOAL1 )) && (( ! sflg )) && (( goal1_strong = spcnt - 1 )) && (( sflg = 1 ))
;;
1) (( wpcnt++ ))
(( wpcnt <= NUM_WEAK )) && weabuff+="${prime.val[j]}, "
(( i >= GOAL1 )) && (( ! wflg )) && (( goal1_weak = wpcnt - 1 )) && (( wflg = 1 ))
;;
99) (( bpcnt++ ))
;;
esac
done
printf "Total primes under %d = %d\n\n" $MAX_INT ${#prime.val[*]}
printf "First %d Strong Primes are: %s\n\n" $NUM_STRONG "${strbuff%,*}"
printf "Number of Strong Primes under %d is: %d\n" $GOAL1 ${goal1_strong}
printf "Number of Strong Primes under %d is: %d\n\n\n" $MAX_INT ${spcnt}
printf "First %d Weak Primes are: %s\n\n" $NUM_WEAK "${weabuff%,*}"
printf "Number of Weak Primes under %d is: %d\n" $GOAL1 ${goal1_weak}
printf "Number of Weak Primes under %d is: %d\n\n\n" $MAX_INT ${wpcnt}
printf "Number of Balanced Primes under %d is: %d\n\n\n" $MAX_INT ${bpcnt}
- Output:
Total primes under 10000000 = 664579
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
Number of Strong Primes under 1000000 is: 37723 Number of Strong Primes under 10000000 is: 320991
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
Number of Weak Primes under 1000000 is: 37779 Number of Weak Primes under 10000000 is: 321749
Number of Balanced Primes under 10000000 is: 21837
Lua
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.
Maple
isStrong := proc(n::posint) local holder;
holder := false;
if isprime(n) and 1/2*prevprime(n) + 1/2*nextprime(n) < n then
holder := true;
end if;
return holder;
end proc:
isWeak := proc(n::posint) local holder;
holder := false;
if isprime(n) and n < 1/2*prevprime(n) + 1/2*nextprime(n) then
holder := true;
end if;
return holder;
end proc
findStrong := proc(n::posint) local count, list, k;
count := 0; list := [];
for k from 3 while count < n do
if isStrong(k) then count := count + 1;
list := [op(list), k];
end if;
end do;
return list;
end proc:
findWeak := proc(n::posint) local count, list, k;
count := 0;
list := [];
for k from 3 while count < n do
if isWeak(k) then
count := count + 1;
list := [op(list), k];
end if;
end do;
return list;
end proc:
findStrong(36)
findWeak(37)
countStrong(1000000)
countStrong(10000000)
countWeak(1000000)
countWeak(10000000)- Output:
[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] [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] 37723 320991 37780 321750
Mathematica /Wolfram Language
p = Prime[Range[PrimePi[10^3]]];
SequenceCases[p, ({a_, b_, c_}) /; (a + c < 2 b) :> b, 36, Overlaps -> True]
SequenceCases[p, ({a_, b_, c_}) /; (a + c > 2 b) :> b, 37, Overlaps -> True]
p = Prime[Range[PrimePi[10^6] + 1]];
Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] < 2 #[[2]] &]]
Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] > 2 #[[2]] &]]
p = Prime[Range[PrimePi[10^7] + 1]];
Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] < 2 #[[2]] &]]
Length[Select[Partition[p, 3, 1], #[[3]] + #[[1]] > 2 #[[2]] &]]
- Output:
{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}
{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}
37723
37780
320991
321750
Nim
import math, strutils
const
M = 10_000_000
N = M + 19 # Maximum value for sieve.
# Fill sieve of Erathosthenes.
var comp: array[2..N, bool] # True means composite; default is prime.
for n in countup(3, sqrt(N.toFloat).int, 2):
if not comp[n]:
for k in countup(n * n, N, 2 * n):
comp[k] = true
# Build list of primes.
var primes = @[2]
for n in countup(3, N, 2):
if not comp[n]:
primes.add n
if primes[^1] < M: quit "Not enough primes: please, increase value of N."
# Build lists of strong and weak primes.
var strongPrimes, weakPrimes: seq[int]
for i in 1..<primes.high:
let p = primes[i]
if p shl 1 > primes[i - 1] + primes[i + 1]:
strongPrimes.add p
elif p shl 1 < primes[i - 1] + primes[i + 1]:
weakPrimes.add p
when isMainModule:
proc count(list: seq[int]; max: int): int =
## Return the count of values less than "max".
for p in list:
if p >= max: break
inc result
echo "First 36 strong primes:"
echo " ", strongPrimes[0..35].join(" ")
echo "Count of strong primes below 1_000_000: ", strongPrimes.count(1_000_000)
echo "Count of strong primes below 10_000_000: ", strongPrimes.count(10_000_000)
echo()
echo "First 37 weak primes:"
echo " ", weakPrimes[0..36].join(" ")
echo "Count of weak primes below 1_000_000: ", weakPrimes.count(1_000_000)
echo "Count of weak primes below 10_000_000: ", weakPrimes.count(10_000_000)
- 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 below 1_000_000: 37723 Count of strong primes below 10_000_000: 320991 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 below 1_000_000: 37780 Count of weak primes below 10_000_000: 321750
Pascal
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
use ntheory qw(primes vecfirst);
sub comma {
(my $s = reverse shift) =~ s/(.{3})/$1,/g;
$s =~ s/,(-?)$/$1/;
$s = reverse $s;
}
sub below { my ($m, @a) = @_; vecfirst { $a[$_] > $m } 0..$#a }
my (@strong, @weak, @balanced);
my @primes = @{ primes(10_000_019) };
for my $k (1 .. $#primes - 1) {
my $x = ($primes[$k - 1] + $primes[$k + 1]) / 2;
if ($x > $primes[$k]) { push @weak, $primes[$k] }
elsif ($x < $primes[$k]) { push @strong, $primes[$k] }
else { push @balanced, $primes[$k] }
}
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 $c1]}: " . comma below($c1,@$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 Count of balanced primes <= 1,000,000: 2,994 Count of balanced primes <= 10,000,000: 21,837
Phix
with javascript_semantics sequence strong = {}, weak = {} for i=2 to get_maxprime(1e14) do -- (ie idx of primes < (sqrt(1e14)==1e7), bar 1st) integer p = get_prime(i), c = compare(p,(get_prime(i-1)+get_prime(i+1))/2) if c=+1 then strong &= p end if if c=-1 then weak &= p end if end for printf(1,"The first thirty six strong primes: %s\n",{join(shorten(strong[1..36],"",4,"%2d"),", ")}) printf(1,"The first thirty seven weak primes: %s\n",{join(shorten( weak[1..37],"",4,"%2d"),", ")}) printf(1,"There are %,d strong primes below %,d and %,d below %,d\n",{abs(binary_search(1e6,strong))-1,1e6,length(strong),1e7}) printf(1,"There are %,d weak primes below %,d and %,d below %,d\n",{abs(binary_search(1e6, weak))-1,1e6,length( weak),1e7})
- Output:
The first thirty six strong primes: 11, 17, 29, 37, ..., 397, 419, 431, 439 The first thirty seven weak primes: 3, 7, 13, 19, ..., 359, 383, 389, 401 There are 37,723 strong primes below 1,000,000 and 320,991 below 10,000,000 There are 37,780 weak primes below 1,000,000 and 321,750 below 10,000,000
PureBasic
#MAX=10000000+20
Global Dim P.b(#MAX) : FillMemory(@P(),#MAX,1,#PB_Byte)
Global NewList Primes.i()
Global NewList Strong.i()
Global NewList Weak.i()
For n=2 To Sqr(#MAX)+1 : If P(n) : m=n*n : While m<=#MAX : P(m)=0 : m+n : Wend : EndIf : Next
For i=2 To #MAX : If p(i) : AddElement(Primes()) : Primes()=i : EndIf : Next
If FirstElement(Primes())
pp=Primes()
While NextElement(Primes())
ap=Primes()
If NextElement(Primes()) : np=Primes() : Else : Break : EndIf
If ap>(pp+np)/2.0 : AddElement(Strong()) : Strong()=ap : If ap<1000000 : c1+1 : EndIf : EndIf
If ap<(pp+np)/2.0 : AddElement(Weak()) : Weak()=ap : If ap<1000000 : c2+1 : EndIf : EndIf
PreviousElement(Primes()) : pp=Primes()
Wend
EndIf
OpenConsole()
If FirstElement(Strong())
PrintN("First 36 strong primes:")
Print(Str(Strong())+" ")
For i=2 To 36 : If NextElement(Strong()) : Print(Str(Strong())+" ") : Else : Break : EndIf : Next
PrintN("")
EndIf
PrintN("Number of strong primes below 1'000'000 = "+FormatNumber(c1,0,".","'"))
PrintN("Number of strong primes below 10'000'000 = "+FormatNumber(ListSize(Strong()),0,".","'"))
If FirstElement(Weak())
PrintN("First 37 weak primes:")
Print(Str(Weak())+" ")
For i=2 To 37 : If NextElement(Weak()) : Print(Str(Weak())+" ") : Else : Break : EndIf : Next
PrintN("")
EndIf
PrintN("Number of weak primes below 1'000'000 = "+FormatNumber(c2,0,".","'"))
PrintN("Number of weak primes below 10'000'000 = "+FormatNumber(ListSize(Weak()),0,".","'"))
Input()
- 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 Number of strong primes below 1'000'000 = 37'723 Number of 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 Number of weak primes below 1'000'000 = 37'780 Number of weak primes below 10'000'000 = 321'750
Python
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.
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
Raku
(formerly Perl 6)
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]».,
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
REXX
/*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: -10000000 weak
321,750 WEAK primes found below or equal to 10,000,000.
Ring
load "stdlib.ring"
see "working..." + nl
p = 0
num = 0
pr1 = 37
pr2 = 38
limit1 = 457
limit2 = 1000000
limit3 = 10000000
primes = []
see "first 36 strong primes:" + nl
while true
p = p + 1
if isprime(p)
if p < limit1
add(primes,p)
else
exit
ok
ok
end
ln = len(primes)
for n = 2 to ln-1
tmp = (primes[n-1] + primes[n+1])/2
if primes[n] > tmp
num = num + 1
if num < pr1
see " " + primes[n]
ok
ok
next
see nl + "first 37 weak primes:" + nl
num = 0
ln = len(primes)
for n = 2 to ln-1
tmp = (primes[n-1] + primes[n+1])/2
if primes[n] < tmp
num = num + 1
if num < pr2
see " " + primes[n]
ok
ok
next
p = 0
primes = []
while true
p = p + 1
if isprime(p)
if p < limit3
add(primes,p)
else
exit
ok
ok
end
primes2 = 0
primes3 = 0
ln = len(primes)
for n = 2 to ln-1
tmp = (primes[n-1] + primes[n+1])/2
if primes[n] > tmp
if primes[n] < limit2
primes2 = primes2 + 1
ok
if primes[n] < limit3
primes3 = primes3 + 1
else
exit
ok
ok
next
see nl
see "strong primes below 1,000,000: " + primes2 + nl
see "strong primes below 10,000,000: " + primes3 + nl
primes2 = 0
primes3 = 0
ln = len(primes)
for n = 2 to ln-1
tmp = (primes[n-1] + primes[n+1])/2
if primes[n] < tmp
if primes[n] < limit2
primes2 = primes2 + 1
ok
if primes[n] < limit3
primes3 = primes3 + 1
else
exit
ok
ok
next
see nl
see "weak primes below 1,000,000: " + primes2 + nl
see "weak primes below 10,000,000: " + primes3 + nlOutput:
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 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 strong primes below 1,000,000: 37723 strong primes below 10,000,000: 320991 weak primes below 1,000,000: 37780 weak primes below 10,000,000: 321750
Ruby
require 'prime'
strong_gen = Enumerator.new{|y| Prime.each_cons(3){|a,b,c|y << b if a+c-b<b} }
weak_gen = Enumerator.new{|y| Prime.each_cons(3){|a,b,c|y << b if a+c-b>b} }
puts "First 36 strong primes:"
puts strong_gen.take(36).join(" "), "\n"
puts "First 37 weak primes:"
puts weak_gen.take(37).join(" "), "\n"
[1_000_000, 10_000_000].each do |limit|
strongs, weaks = 0, 0
Prime.each_cons(3) do |a,b,c|
strongs += 1 if b > a+c-b
weaks += 1 if b < a+c-b
break if c > limit
end
puts "#{strongs} strong primes and #{weaks} weak primes below #{limit}."
end
- 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 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 37723 strong primes and 37780 weak primes below 1000000. 320991 strong primes and 321750 weak primes below 10000000.
Rust
fn is_prime(n: i32) -> bool {
for i in 2..n {
if i * i > n {
return true;
}
if n % i == 0 {
return false;
}
}
n > 1
}
fn next_prime(n: i32) -> i32 {
for i in (n+1).. {
if is_prime(i) {
return i;
}
}
0
}
fn main() {
let mut n = 0;
let mut prime_q = 5;
let mut prime_p = 3;
let mut prime_o = 2;
print!("First 36 strong primes: ");
while n < 36 {
if prime_p > (prime_o + prime_q) / 2 {
print!("{} ",prime_p);
n += 1;
}
prime_o = prime_p;
prime_p = prime_q;
prime_q = next_prime(prime_q);
}
println!("");
while prime_p < 1000000 {
if prime_p > (prime_o + prime_q) / 2 {
n += 1;
}
prime_o = prime_p;
prime_p = prime_q;
prime_q = next_prime(prime_q);
}
println!("strong primes below 1,000,000: {}", n);
while prime_p < 10000000 {
if prime_p > (prime_o + prime_q) / 2 {
n += 1;
}
prime_o = prime_p;
prime_p = prime_q;
prime_q = next_prime(prime_q);
}
println!("strong primes below 10,000,000: {}", n);
n = 0;
prime_q = 5;
prime_p = 3;
prime_o = 2;
print!("First 36 weak primes: ");
while n < 36 {
if prime_p < (prime_o + prime_q) / 2 {
print!("{} ",prime_p);
n += 1;
}
prime_o = prime_p;
prime_p = prime_q;
prime_q = next_prime(prime_q);
}
println!("");
while prime_p < 1000000 {
if prime_p < (prime_o + prime_q) / 2 {
n += 1;
}
prime_o = prime_p;
prime_p = prime_q;
prime_q = next_prime(prime_q);
}
println!("weak primes below 1,000,000: {}", n);
while prime_p < 10000000 {
if prime_p < (prime_o + prime_q) / 2 {
n += 1;
}
prime_o = prime_p;
prime_p = prime_q;
prime_q = next_prime(prime_q);
}
println!("weak primes below 10,000,000: {}", n);
}
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: 37723 strong primes below 10,000,000: 320991 First 36 weak primes: 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: 37779 weak primes below 10,000,000: 321749
Scala
This example works entirely with lazily evaluated lists. It starts with a list of primes, and generates a sliding iterator that looks at each triplet of primes. Lists of strong and weak primes are built by applying the given filters then selecting the middle term from each triplet.
object StrongWeakPrimes {
def main(args: Array[String]): Unit = {
val bnd = 1000000
println(
f"""|First 36 Strong Primes: ${strongPrimes.take(36).map(n => f"$n%,d").mkString(", ")}
|Strong Primes < 1,000,000: ${strongPrimes.takeWhile(_ < bnd).size}%,d
|Strong Primes < 10,000,000: ${strongPrimes.takeWhile(_ < 10*bnd).size}%,d
|
|First 37 Weak Primes: ${weakPrimes.take(37).map(n => f"$n%,d").mkString(", ")}
|Weak Primes < 1,000,000: ${weakPrimes.takeWhile(_ < bnd).size}%,d
|Weak Primes < 10,000,000: ${weakPrimes.takeWhile(_ < 10*bnd).size}%,d""".stripMargin)
}
def weakPrimes: LazyList[Int] = primeTrips.filter{case a +: b +: c +: _ => b < (a + c)/2.0}.map(_(1)).to(LazyList)
def strongPrimes: LazyList[Int] = primeTrips.filter{case a +: b +: c +: _ => b > (a + c)/2}.map(_(1)).to(LazyList)
def primeTrips: Iterator[LazyList[Int]] = primes.sliding(3)
def primes: LazyList[Int] = 2 #:: LazyList.from(3, 2).filter(n => !Iterator.range(3, math.sqrt(n).toInt + 1, 2).exists(n%_ == 0))
}
- 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 < 1,000,000: 37,723 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 Weak Primes < 1,000,000: 37,780 Weak Primes < 10,000,000: 321,750
Sidef
var primes = 10_000_019.primes
var (*strong, *weak, *balanced)
for k in (1 ..^ primes.end) {
var p = primes[k]
given((primes[k-1] + primes[k+1])/2) { |x|
case (x > p) { weak << p }
case (x < p) { strong << p }
else { balanced << p }
}
}
for pr, type, d, c1, c2 in [
[ strong, 'strong', 36, 1e6, 1e7],
[ weak, 'weak', 37, 1e6, 1e7],
[balanced, 'balanced', 28, 1e6, 1e7],
] {
say ("\nFirst #{d} #{type} primes:\n", pr.first(d).map{.commify}.join(' '))
say ("Count of #{type} primes <= #{c1.commify}: ", pr.first_index { _ > 1e6 }.commify)
say ("Count of #{type} primes <= #{c2.commify}: " , pr.len.commify)
}
- 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
Swift
import Foundation
class PrimeSieve {
var composite: [Bool]
init(size: Int) {
composite = Array(repeating: false, count: size/2)
var p = 3
while p * p <= size {
if !composite[p/2 - 1] {
let inc = p * 2
var q = p * p
while q <= size {
composite[q/2 - 1] = true
q += inc
}
}
p += 2
}
}
func isPrime(number: Int) -> Bool {
if number < 2 {
return false
}
if (number & 1) == 0 {
return number == 2
}
return !composite[number/2 - 1]
}
}
func commatize(_ number: Int) -> String {
let n = NSNumber(value: number)
return NumberFormatter.localizedString(from: n, number: .decimal)
}
let limit1 = 1000000
let limit2 = 10000000
class PrimeInfo {
let maxPrint: Int
var count1: Int
var count2: Int
var primes: [Int]
init(maxPrint: Int) {
self.maxPrint = maxPrint
count1 = 0
count2 = 0
primes = []
}
func addPrime(prime: Int) {
count2 += 1
if prime < limit1 {
count1 += 1
}
if count2 <= maxPrint {
primes.append(prime)
}
}
func printInfo(name: String) {
print("First \(maxPrint) \(name) primes: \(primes)")
print("Number of \(name) primes below \(commatize(limit1)): \(commatize(count1))")
print("Number of \(name) primes below \(commatize(limit2)): \(commatize(count2))")
}
}
var strongPrimes = PrimeInfo(maxPrint: 36)
var weakPrimes = PrimeInfo(maxPrint: 37)
let sieve = PrimeSieve(size: limit2 + 100)
var p1 = 2, p2 = 3, p3 = 5
while p2 < limit2 {
if sieve.isPrime(number: p3) {
let diff = p1 + p3 - 2 * p2
if diff < 0 {
strongPrimes.addPrime(prime: p2)
} else if diff > 0 {
weakPrimes.addPrime(prime: p2)
}
p1 = p2
p2 = p3
}
p3 += 2
}
strongPrimes.printInfo(name: "strong")
weakPrimes.printInfo(name: "weak")
- 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] Number of strong primes below 1,000,000: 37,723 Number of 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] Number of weak primes below 1,000,000: 37,780 Number of weak primes below 10,000,000: 321,750
Wren
import "./math" for Int
import "./fmt" for Fmt
var primes = Int.primeSieve(1e7 + 19) // next prime above 10 million
var strong = []
var weak = []
for (p in 1...primes.count-1) {
if (primes[p] > (primes[p-1] + primes[p+1]) / 2) {
strong.add(primes[p])
} else if (primes[p] < (primes[p-1] + primes[p+1]) / 2) {
weak.add(primes[p])
}
}
System.print("The first 36 strong primes are:")
Fmt.print("$d", strong.take(36))
Fmt.print("\nThe count of the strong primes below $,d is $,d.", 1e6, strong.count{ |n| n < 1e6 })
Fmt.print("\nThe count of the strong primes below $,d is $,d.", 1e7, strong.count{ |n| n < 1e7 })
System.print("\nThe first 37 weak primes are:")
Fmt.print("$d", weak.take(37))
Fmt.print("\nThe count of the weak primes below $,d is $,d.", 1e6, weak.count{ |n| n < 1e6 })
Fmt.print("\nThe count of the weak primes below $,d is $,d.", 1e7, weak.count{ |n| n < 1e7 })
- 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 count of the strong primes below 1,000,000 is 37,723. The count of the 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 count of the weak primes below 1,000,000 is 37,780. The count of the weak primes below 10,000,000 is 321,750.
XPL0
proc NumOut(Num); \Output positive integer with commas
int Num, Dig, Cnt;
[Cnt:= [0];
Num:= Num/10;
Dig:= rem(0);
Cnt(0):= Cnt(0)+1;
if Num then NumOut(Num);
Cnt(0):= Cnt(0)-1;
ChOut(0, Dig+^0);
if rem(Cnt(0)/3)=0 & Cnt(0) then ChOut(0, ^,);
];
func IsPrime(N); \Return 'true' if odd N > 2 is prime
int N, I;
[for I:= 3 to sqrt(N) do
[if rem(N/I) = 0 then return false;
I:= I+1;
];
return true;
];
int StrongCnt, WeakCnt, StrongCnt0, WeakCnt0, Strongs(36), Weaks(37);
int N, P0, P1, P2, T;
[StrongCnt:= 0; WeakCnt:= 1;
Weaks(0):= 3;
N:= 7; P1:= 3; P2:= 5; \handles unique case where (2+5)/2 = 3.5
repeat if IsPrime(N) then
[P0:= P1; P1:= P2; P2:= N;
T:= (P0+P2)/2;
if P1 > T then
[if StrongCnt < 36 then Strongs(StrongCnt):= P1;
StrongCnt:= StrongCnt+1;
];
if P1 < T then
[if WeakCnt < 37 then Weaks(WeakCnt):= P1;
WeakCnt:= WeakCnt+1;
];
];
if P1 < 1_000_000 then
[StrongCnt0:= StrongCnt; WeakCnt0:= WeakCnt];
N:= N+2;
until P1 >= 10_000_000;
Text(0, "First 36 strong primes:^M^J");
for N:= 0 to 36-1 do
[NumOut(Strongs(N)); ChOut(0, ^ )];
Text(0, "^M^JStrong primes below 1,000,000: ");
NumOut(StrongCnt0);
Text(0, "^M^JStrong primes below 10,000,000: ");
NumOut(StrongCnt);
Text(0, "^M^JFirst 37 weak primes:^M^J");
for N:= 0 to 37-1 do
[NumOut(Weaks(N)); ChOut(0, ^ )];
Text(0, "^M^JWeak primes below 1,000,000: ");
NumOut(WeakCnt0);
Text(0, "^M^JWeak primes below 10,000,000: ");
NumOut(WeakCnt);
CrLf(0);
]- 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: 321,751
zkl
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