Successive prime differences: Difference between revisions

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The series of increasing prime numbers begins: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

The task applies a filter to the series returning groups of successive primes, (s'primes), that differ from the next by a given value or values.

Example 1: Specifying that the difference between s'primes be 2 leads to the groups:

(3, 5), (5, 7), (11, 13), (17, 19), (29, 31), ...

(Known as Twin primes or Prime pairs)

Example 2: Specifying more than one difference between s'primes leads to groups of size one greater than the number of differences. Differences of 2, 4 leads to the groups:

(5, 7, 11), (11, 13, 17), (17, 19, 23), (41, 43, 47), ....

In the first group 7 is two more than 5 and 11 is four more than 7; as well as 5, 7, and 11 being successive primes. Differences are checked in the order of the values given, (differences of 4, 2 would give different groups entirely).

Task

In each case use a list of primes less than 1_000_000
For the following Differences show the first and last group, as well as the number of groups found:

Differences of 2.
Differences of 1.
Differences of 2, 2.
Differences of 2, 4.
Differences of 4, 2.
Differences of 6, 4, 2.

Show output here.

Note: Generation of a list of primes is a secondary aspect of the task. Use of a built in function, well known library, or importing/use of prime generators from other Rosetta Code tasks is encouraged.

references

11l

Translation of: D

<lang 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)

F successive_primes(primes, diffs)

  Int results
  V dl = diffs.len

  L(i) 0 .< primes.len - dl
     V group = [0] * (dl + 1)
     group[0] = primes[i]
     L(j) i .+ dl
        I primes[j + 1] - primes[j] != diffs[j - i]
           L.break
        group[j - i + 1] = primes[j + 1]
     L.was_no_break
        results [+]= group

  R results

V prime_list = primes_upto(1'000'000)

print(‘For primes less than 1,000,000:-’) L(diffs) [[2], [1], [2, 2], [2, 4], [4, 2], [6, 4, 2]]

  print(‘  For differences of #. ->’.format(diffs))
  V sp = successive_primes(prime_list, diffs)
  print(‘    First group   = ’sp[0])
  print(‘    Last group    = ’sp.last)
  print(‘    Number found  = ’sp.len)
  print()</lang>

Output:

For primes less than 1,000,000:-
  For differences of [2] ->
    First group   = [3, 5]
    Last group    = [999959, 999961]
    Number found  = 8169

  For differences of [1] ->
    First group   = [2, 3]
    Last group    = [2, 3]
    Number found  = 1

  For differences of [2, 2] ->
    First group   = [3, 5, 7]
    Last group    = [3, 5, 7]
    Number found  = 1

  For differences of [2, 4] ->
    First group   = [5, 7, 11]
    Last group    = [999431, 999433, 999437]
    Number found  = 1393

  For differences of [4, 2] ->
    First group   = [7, 11, 13]
    Last group    = [997807, 997811, 997813]
    Number found  = 1444

  For differences of [6, 4, 2] ->
    First group   = [31, 37, 41, 43]
    Last group    = [997141, 997147, 997151, 997153]
    Number found  = 306

ALGOL 68

<lang algol68>BEGIN # find some sequences of primes where the gaps between the elements #

     # follow specific patterns                                          #
   # reurns a list of primes up to n #
   PROC prime list = ( INT n )[]INT:
        BEGIN
           # sieve the primes to n #
           INT no = 0, yes = 1;
           [ 1 : n ]INT p;
           p[ 1 ] := no; p[ 2 ] := yes;
           FOR i FROM 3 BY 2 TO n DO p[ i ] := yes OD;
           FOR i FROM 4 BY 2 TO n DO p[ i ] := no  OD;
           FOR i FROM 3 BY 2 TO ENTIER sqrt( n ) DO
               IF p[ i ] = yes THEN FOR s FROM i * i BY i + i TO n DO p[ s ] := no OD FI
           OD;
           # replace the sieve with a list #
           INT p pos := 0;
           FOR i TO n DO IF p[ i ] = yes THEN p[ p pos +:= 1 ] := i FI OD;
           p[ 1 : p pos ]
        END # prime list # ;
   # prints the elements of list #
   PROC print list = ( STRING name, []INT list )VOID:
        BEGIN
           print( ( name, "[" ) );
           FOR i FROM LWB list TO UPB list DO print( ( " ", whole( list[ i ], 0 ) ) ) OD;
           print( ( " ]" ) )
        END # print list # ;
   # attempts to find patterns in the differences of primes and prints the results #
   PROC try differences = ( []INT primes, []INT pattern )VOID:
        BEGIN
           INT    pattern length = ( UPB pattern - LWB pattern ) + 1;
           [ 1 :  pattern length + 1 ]INT first; FOR i TO UPB first DO first[ i ] := 0 OD;
           [ 1 :  pattern length + 1 ]INT last;  FOR i TO UPB last  DO last[  i ] := 0 OD;
           INT    count := 0;
           FOR p FROM LWB primes + pattern length TO UPB primes DO
               BOOL matched := TRUE;
               INT e pos    := LWB pattern;
               FOR e FROM p - pattern length TO p - 1
               WHILE matched := primes[ e + 1 ] - primes[ e ] = pattern[ e pos ]
               DO
                   e pos +:= 1
               OD;
               IF matched THEN
                   # found a matching sequence #
                   count +:= 1;
                   last   := primes[ p - pattern length : p @ 1 ];
                   IF count = 1 THEN first := last FI
               FI
           OD;
           print( ( "  Found ", whole( count, 0 ), " prime sequence(s) that differ by: " ) );
           print list( "", pattern );
           print( ( newline ) );
           IF count > 0 THEN
               # found at least one sequence #
               print list( "    first: ", first );
               print list( "    last: ",  last  );
               print( ( newline ) )
           FI;
           print( ( newline ) )
        END # try differences # ;
   INT max number = 1 000 000;
   []INT    p list = prime list( max number );
   print( ( "For primes up to ", whole( max number, 0 ), "...", newline ) );
   try differences( p list, ( 2 )               );try differences( p list, ( 1 )               );
   try differences( p list, ( 2, 2 )            );try differences( p list, ( 2, 4 )            );
   try differences( p list, ( 4, 2 )            );try differences( p list, ( 6, 4, 2 )         );
   try differences( p list, ( 2, 4, 6, 8 )      );try differences( p list, ( 2, 4, 6, 8, 10 )  );
   try differences( p list, ( 32, 16, 8, 4, 2 ) )

END</lang>

Output:

For primes up to 1000000...
  Found 8169 prime sequence(s) that differ by: [ 2 ]
    first: [ 3 5 ]    last: [ 999959 999961 ]

  Found 1 prime sequence(s) that differ by: [ 1 ]
    first: [ 2 3 ]    last: [ 2 3 ]

  Found 1 prime sequence(s) that differ by: [ 2 2 ]
    first: [ 3 5 7 ]    last: [ 3 5 7 ]

  Found 1393 prime sequence(s) that differ by: [ 2 4 ]
    first: [ 5 7 11 ]    last: [ 999431 999433 999437 ]

  Found 1444 prime sequence(s) that differ by: [ 4 2 ]
    first: [ 7 11 13 ]    last: [ 997807 997811 997813 ]

  Found 306 prime sequence(s) that differ by: [ 6 4 2 ]
    first: [ 31 37 41 43 ]    last: [ 997141 997147 997151 997153 ]

  Found 68 prime sequence(s) that differ by: [ 2 4 6 8 ]
    first: [ 347 349 353 359 367 ]    last: [ 984911 984913 984917 984923 984931 ]

  Found 11 prime sequence(s) that differ by: [ 2 4 6 8 10 ]
    first: [ 13901 13903 13907 13913 13921 13931 ]    last: [ 954257 954259 954263 954269 954277 954287 ]

  Found 1 prime sequence(s) that differ by: [ 32 16 8 4 2 ]
    first: [ 148091 148123 148139 148147 148151 148153 ]    last: [ 148091 148123 148139 148147 148151 148153 ]

AWK

syntax: GAWK -f SUCCESSIVE_PRIME_DIFFERENCES.AWK

BEGIN {

   for (i=lo=0; i<=hi=1000000; i++) {
     if (is_prime(i)) {
       p_arr[++p] = i
     }
   }
   printf("there are %d primes between %d - %d\n",p,lo,hi)
   fmt = "%-11s %5s %-15s %s\n"
   printf(fmt,"differences","count","first group","last group")
   for (a=1; a<=split("2;1;2,2;2,4;4,2;6,4,2;2,4,6;100;112",diff_arr,";"); a++) {
     diff_leng = split(diff_arr[a],tmp_arr,",")
     first_set = last_set = ""
     count = 0
     for (b=1; b<=p; b++) {
       str = ""
       for (c=1; c<=diff_leng; c++) {
         if (p_arr[b+c-1] + tmp_arr[c] == p_arr[b+c]) {
           str = (str == "") ? (p_arr[b+c-1] "," p_arr[b+c]) : (str "," p_arr[b+c])
         }
       }
       if (gsub(/,/,"&",str) == diff_leng) {
         count++
         if (first_set == "") {
           first_set = str
         }
         last_set = str
       }
     }
     printf(fmt,diff_arr[a],count,first_set,last_set)
   }
   exit(0)

} function is_prime(x, i) {

   if (x <= 1) {
     return(0)
   }
   for (i=2; i<=int(sqrt(x)); i++) {
     if (x % i == 0) {
       return(0)
     }
   }
   return(1)

} </lang>

Output:

there are 78498 primes between 0 - 1000000
differences count first group     last group
2            8169 3,5             999959,999961
1               1 2,3             2,3
2,2             1 3,5,7           3,5,7
2,4          1393 5,7,11          999431,999433,999437
4,2          1444 7,11,13         997807,997811,997813
6,4,2         306 31,37,41,43     997141,997147,997151,997153
2,4,6         279 17,19,23,29     997097,997099,997103,997109
100             2 396733,396833   838249,838349
112             1 370261,370373   370261,370373

C

<lang c>#include <stdbool.h>

include <stdint.h>
include <stdio.h>

define PRIME_COUNT 100000

int64_t PRIMES[PRIME_COUNT]; size_t primeSize = 0;

bool isPrime(int n) {

   size_t i = 0;

   for (i = 0; i < primeSize; i++) {
       int64_t p = PRIMES[i];
       if (n == p) {
           return true;
       }
       if (n % p == 0) {
           return false;
       }
       if (p * p > n) {
           break;
       }
   }

   return true;

}

void initialize() {

   int i;

   PRIMES[primeSize++] = 2;
   PRIMES[primeSize++] = 3;
   PRIMES[primeSize++] = 5;
   PRIMES[primeSize++] = 7;
   PRIMES[primeSize++] = 11;
   PRIMES[primeSize++] = 13;
   PRIMES[primeSize++] = 17;
   PRIMES[primeSize++] = 19;

   for (i = 21; primeSize < PRIME_COUNT;) {
       if (isPrime(i)) {
           PRIMES[primeSize++] = i;
       }
       i += 2;

       if (primeSize < PRIME_COUNT && isPrime(i)) {
           PRIMES[primeSize++] = i;
       }
       i += 2;
   }

}

void diff1(size_t diff) {

   int64_t pm0, pm1;
   int64_t fg1 = 0, fg2 = 0, lg1 = 0, lg2 = 0;
   size_t pos, count = 0;

   if (diff == 0) {
       return;
   }

   pm0 = PRIMES[0];
   for (pos = 1; pos < PRIME_COUNT; pos++) {
       pm1 = pm0;
       pm0 = PRIMES[pos];
       if (pm0 > 1000000) {
           break;
       }
       if (pm0 - pm1 == diff) {
           count++;
           if (fg1 == 0) {
               fg1 = pm1;
               fg2 = pm0;
           }
           lg1 = pm1;
           lg2 = pm0;
       }
   }

   printf("%ld|%d|%lld %lld|%lld %lld|\n", diff, count, fg1, fg2, lg1, lg2);

}

void diff2(size_t d0, size_t d1) {

   int64_t pm0, pm1, pm2;
   int64_t fg1 = 0, fg2, fg3, lg1, lg2, lg3;
   size_t pos, count = 0;

   if (d0 == 0 || d1 == 0) {
       return;
   }

   pm1 = PRIMES[0];
   pm0 = PRIMES[1];
   for (pos = 2; pos < PRIME_COUNT; pos++) {
       pm2 = pm1;
       pm1 = pm0;
       pm0 = PRIMES[pos];
       if (pm0 > 1000000) {
           break;
       }
       if (pm1 - pm2 == d0 && pm0 - pm1 == d1) {
           count++;
           if (fg1 == 0) {
               fg1 = pm2;
               fg2 = pm1;
               fg3 = pm0;
           }
           lg1 = pm2;
           lg2 = pm1;
           lg3 = pm0;
       }
   }

   printf("%d %d|%d|%lld %lld %lld|%lld %lld %lld|\n", d0, d1, count, fg1, fg2, fg3, lg1, lg2, lg3);

}

void diff3(size_t d0, size_t d1, size_t d2) {

   int64_t pm0, pm1, pm2, pm3;
   int64_t fg1 = 0, fg2, fg3, fg4, lg1, lg2, lg3, lg4;
   size_t pos, count = 0;

   if (d0 == 0 || d1 == 0 || d2 == 0) {
       return;
   }

   pm2 = PRIMES[0];
   pm1 = PRIMES[1];
   pm0 = PRIMES[2];
   for (pos = 3; pos < PRIME_COUNT; pos++) {
       pm3 = pm2;
       pm2 = pm1;
       pm1 = pm0;
       pm0 = PRIMES[pos];
       if (pm0 > 1000000) {
           break;
       }
       if (pm2 - pm3 == d0 && pm1 - pm2 == d1 && pm0 - pm1 == d2) {
           count++;
           if (fg1 == 0) {
               fg1 = pm3;
               fg2 = pm2;
               fg3 = pm1;
               fg4 = pm0;
           }
           lg1 = pm3;
           lg2 = pm2;
           lg3 = pm1;
           lg4 = pm0;
       }
   }

   printf("%d %d %d|%d|%lld %lld %lld %lld|%lld %lld %lld %lld|\n", d0, d1, d2, count, fg1, fg2, fg3, fg4, lg1, lg2, lg3, lg4);

}

int main() {

   initialize();

   printf("differences|count|first group|last group\n");

   diff1(2);
   diff1(1);

   diff2(2, 2);
   diff2(2, 4);
   diff2(4, 2);

   diff3(6, 4, 2);

   return 0;

}</lang>

Output:

differences|count|first group|last group
2|8169|3 5|999959 999961|
1|1|2 3|2 3|
2 2|1|3 5 7|3 5 7|
2 4|1393|5 7 11|999431 999433 999437|
4 2|1444|7 11 13|997807 997811 997813|
6 4 2|306|31 37 41 43|997141 997147 997151 997153|

C#

<lang csharp>using System; using System.Collections.Generic; using static System.Linq.Enumerable;

public static class SuccessivePrimeDifferences {

   public static void Main() {
       var primes = GeneratePrimes(1_000_000).ToList();
       foreach (var d in new[] {
           new [] { 2 },
           new [] { 1 },
           new [] { 2, 2 },
           new [] { 2, 4 },
           new [] { 4, 2 },
           new [] { 6, 4, 2 },
       }) {
           IEnumerable<int> first = null, last = null;
           int count = 0;
           foreach (var grp in FindDifferenceGroups(d)) {
               if (first == null) first = grp;
               last = grp;
               count++;
           }
           Console.WriteLine($"{$"({string.Join(", ", first)})"}, {$"({string.Join(", ", last)})"}, {count}");
       }

       IEnumerable<IEnumerable<int>> FindDifferenceGroups(int[] diffs) {
           for (int pi = diffs.Length; pi < primes.Count; pi++)
               if (Range(0, diffs.Length).All(di => primes[pi-diffs.Length+di+1] - primes[pi-diffs.Length+di] == diffs[di]))
                   yield return Range(pi - diffs.Length, diffs.Length + 1).Select(i => primes[i]);
       }

       IEnumerable<int> GeneratePrimes(int lmt) {
           bool[] comps = new bool[lmt + 1];
           comps[0] = comps[1] = true;
           yield return 2; yield return 3;
           for (int j = 4; j <= lmt; j += 2) comps[j] = true;
           for (int j = 9; j <= lmt; j += 6) comps[j] = true;
           int i = 5, d = 4, rt = (int)Math.Sqrt(lmt);
           for ( ; i <= rt; i += (d = 6 - d))
               if (!comps[i]) {
                   yield return i;
                   for (int j = i * i, k = i << 1; j <= lmt; j += k)
                       comps[j] = true;
               }
           for ( ; i <= lmt; i += (d = 6 - d)) if (!comps[i]) yield return i;
       }
   }

}</lang>

Output:

(3, 5), (999959, 999961), 8169
(2, 3), (2, 3), 1
(3, 5, 7), (3, 5, 7), 1
(5, 7, 11), (999431, 999433, 999437), 1393
(7, 11, 13), (997807, 997811, 997813), 1444
(31, 37, 41, 43), (997141, 997147, 997151, 997153), 306

C++

<lang cpp>#include <iostream>

include <cstdint>
include <vector>
include "prime_sieve.hpp"

using integer = uint32_t; using vector = std::vector<integer>;

void print_vector(const vector& vec) {

   if (!vec.empty()) {
       auto i = vec.begin();
       std::cout << '(' << *i;
       for (++i; i != vec.end(); ++i)
           std::cout << ", " << *i;
       std::cout << ')';
   }

}

class diffs { public:

   diffs(std::initializer_list<integer> list) : diffs_(list) {}
   diffs(const vector& vec) : diffs_(vec) {}
   void test_group(const vector&);
   size_t size() const {
       return diffs_.size();
   }
   void print(std::ostream&);

private:

   vector diffs_;
   vector first_;
   vector last_;
   integer count_ = 0;

};

void diffs::test_group(const vector& vec) {

   if (vec.size() < size() + 1)
       return;
   size_t start = vec.size() - size() - 1;
   for (size_t i = 0, j = start + 1; i < size(); ++i, ++j) {
       if (vec[j] - vec[j - 1] != diffs_[i])
           return;
   }
   vector group(vec.begin() + start, vec.end());
   if (count_ == 0)
       first_ = group;
   last_ = group;
   ++count_;

}

void diffs::print(std::ostream& out) {

   print_vector(diffs_);
   out << ": first group = ";
   print_vector(first_);
   out << ", last group = ";
   print_vector(last_);
   out << ", count = " << count_ << '\n';

}

int main() {

   const integer limit = 1000000;
   const size_t max_group_size = 4;
   prime_sieve sieve(limit);
   diffs d[] = { {2}, {1}, {2, 2}, {2, 4}, {4, 2}, {6, 4, 2} };
   vector group;
   for (integer p = 0; p < limit; ++p) {
       if (!sieve.is_prime(p))
           continue;
       if (group.size() >= max_group_size)
           group.erase(group.begin());
       group.push_back(p);
       for (auto&& diff : d) {
           diff.test_group(group);
       }
   }
   for (auto&& diff : d) {
       diff.print(std::cout);
   }
   return 0;

}</lang>

Contents of prime_sieve.hpp: <lang cpp>#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</lang>

Output:

(2): first group = (3, 5), last group = (999959, 999961), count = 8169
(1): first group = (2, 3), last group = (2, 3), count = 1
(2, 2): first group = (3, 5, 7), last group = (3, 5, 7), count = 1
(2, 4): first group = (5, 7, 11), last group = (999431, 999433, 999437), count = 1393
(4, 2): first group = (7, 11, 13), last group = (997807, 997811, 997813), count = 1444
(6, 4, 2): first group = (31, 37, 41, 43), last group = (997141, 997147, 997151, 997153), count = 306

D

Translation of: Go

<lang 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;

}

int[][] successivePrimes(int[] primes, int[] diffs) {

   int[][] results;
   auto dl = diffs.length;

   outer:
   for (int i = 0; i < primes.length - dl; i++) {
       auto group = uninitializedArray!(int[])(dl + 1);
       group[0] = primes[i];
       for (int j = i; j < i + dl; j++) {
           if (primes[j + 1] - primes[j] != diffs[j - i]) {
               continue outer;
           }
           group[j - i + 1] = primes[j + 1];
       }
       results ~= group;
   }

   return results;

}

void main() {

   auto primeList = iota(2, 1_000_000).filter!isPrime.array;
   auto diffsList = [[2], [1], [2, 2], [2, 4], [4, 2], [6, 4, 2]];
   writeln("For primes less than 1,000,000:-");
   foreach (diffs; diffsList) {
       writefln("  For differences of %s ->", diffs);
       auto sp = successivePrimes(primeList, diffs);
       if (sp.length == 0) {
           writeln("    No groups found");
           continue;
       }
       writeln("    First group   = ", sp[0]);
       writeln("    Last group    = ", sp[$ - 1]);
       writeln("    Number found  = ", sp.length);
       writeln();
   }

}</lang>

Output:

For primes less than 1,000,000:-
  For differences of [2] ->
    First group   = [3, 5]
    Last group    = [999959, 999961]
    Number found  = 8169

  For differences of [1] ->
    First group   = [2, 3]
    Last group    = [2, 3]
    Number found  = 1

  For differences of [2, 2] ->
    First group   = [3, 5, 7]
    Last group    = [3, 5, 7]
    Number found  = 1

  For differences of [2, 4] ->
    First group   = [5, 7, 11]
    Last group    = [999431, 999433, 999437]
    Number found  = 1393

  For differences of [4, 2] ->
    First group   = [7, 11, 13]
    Last group    = [997807, 997811, 997813]
    Number found  = 1444

  For differences of [6, 4, 2] ->
    First group   = [31, 37, 41, 43]
    Last group    = [997141, 997147, 997151, 997153]
    Number found  = 306

Delphi

Library: System.SysUtils

Library: System.Generics.Collections

<lang Delphi> program Successive_prime_differences;

{$APPTYPE CONSOLE}

{$R *.res}

uses

 System.SysUtils,
 System.Generics.Collections;

function IsPrime(a: UInt64): Boolean; var

 d: UInt64;

begin

 if (a < 2) then
   exit(False);

 if (a mod 2) = 0 then
   exit(a = 2);

 if (a mod 3) = 0 then
   exit(a = 3);

 d := 5;

 while (d * d <= a) do
 begin
   if (a mod d = 0) then
     Exit(false);
   inc(d, 2);

   if (a mod d = 0) then
     Exit(false);
   inc(d, 4);
 end;

 Result := True;

end;

function Primes(const limit: UInt64): TArray<UInt64>; var

 i: UInt64;

 procedure Add(const value: UInt64);
 begin
   SetLength(result, Length(result) + 1);
   Result[Length(result) - 1] := value;
 end;

begin

 if limit < 2 then
   exit;

 // 2 is the only even prime
 Add(2);

 i := 3;
 while i <= limit do
 begin
   if IsPrime(i) then
     Add(i);
   inc(i, 2);
 end;

end;

function Commatize(const n: UInt64): string; var

 str: string;
 digits: Integer;
 i: Integer;

begin

 Result := ;
 str := n.ToString;
 digits := str.Length;

 for i := 1 to digits do
 begin
   if ((i > 1) and (((i - 1) mod 3) = (digits mod 3))) then
     Result := Result + ',';
   Result := Result + str[i];
 end;

end;

function CheckScan(index: Integer; p: TArray<UInt64>; pattern: array of Integer): Boolean; var

 i, last: Integer;

begin

 last := Length(pattern) - 1;
 for i := 0 to last do
   if p[index - last + i - 1] + pattern[i] <> p[index - last + i] then
     exit(False);
 Result := True;

end;

const

 GroupLabel: array[1..6] of string = ('(2)', '(1)', '(2, 2)', '(2, 4)',
   '(4, 2)', '(6, 4, 2)');

var

 limit, start: UInt64;
 c: TArray<UInt64>;
 i, j: UInt64;
 Group: array[1..6] of Tlist<string>;

begin

 for i := 1 to 6 do
   Group[i] := Tlist<string>.Create;

 limit := Trunc(1e6 - 1);
 c := Primes(limit);

 for j := 1 to High(c) do
 begin
   if CheckScan(j, c, [2]) then
     Group[1].Add(format('(%d,%d)', [c[j - 1], c[j]]));

   if CheckScan(j, c, [1]) then
     Group[2].Add(format('(%d,%d)', [c[j - 1], c[j]]));

   if j > 1 then
   begin
     if CheckScan(j, c, [2, 2]) then
       Group[3].Add(format('(%d,%d,%d)', [c[j - 2], c[j - 1], c[j]]));

     if CheckScan(j, c, [2, 4]) then
       Group[4].Add(format('(%d,%d,%d)', [c[j - 2], c[j - 1], c[j]]));

     if CheckScan(j, c, [4, 2]) then
       Group[5].Add(format('(%d,%d,%d)', [c[j - 2], c[j - 1], c[j]]));
   end;

   if j > 2 then
     if CheckScan(j, c, [6, 4, 2]) then
       Group[6].Add(format('(%d,%d,%d,%d)', [c[j - 3], c[j - 2], c[j - 1], c[j]]));

 end;

 for i := 1 to 6 do
 begin
   Write(GroupLabel[i], ': first group = ', Group[i].First);
   Writeln(', last group = ', Group[i].last, ', count = ', Group[i].Count);
   Group[i].free;
 end;

 readln;

end.

</lang>

Output:

(2): first group = (3,5), last group = (999959,999961), count = 8169
(1): first group = (2,3), last group = (2,3), count = 1
(2, 2): first group = (3,5,7), last group = (3,5,7), count = 1
(2, 4): first group = (5,7,11), last group = (999431,999433,999437), count = 1393
(4, 2): first group = (7,11,13), last group = (997807,997811,997813), count = 1444
(6, 4, 2): first group = (31,37,41,43), last group = (997141,997147,997151,997153), count = 306

F#

This task uses Extensible Prime Generator (F#) <lang fsharp> // Successive primes. Nigel Galloway: May 6th., 2019 let sP n=let sP=pCache|>Seq.takeWhile(fun n->n<1000000)|>Seq.windowed(Array.length n+1)|>Seq.filter(fun g->g=(Array.scan(fun n g->n+g) g.[0] n))

        printfn "sP %A\t-> Min element = %A Max element = %A of %d elements" n (Seq.head sP) (Seq.last sP) (Seq.length sP)

List.iter sP [[|2|];[|1|];[|2;2|];[|2;4|];[|4;2|];[|6;4;2|]] </lang>

Output:

sP [|2|]        -> Min element = [|3; 5|] Max element = [|999959; 999961|] of 8169 elements
sP [|1|]        -> Min element = [|2; 3|] Max element = [|2; 3|] of 1 elements
sP [|2; 2|]     -> Min element = [|3; 5; 7|] Max element = [|3; 5; 7|] of 1 elements
sP [|2; 4|]     -> Min element = [|5; 7; 11|] Max element = [|999431; 999433; 999437|] of 1393 elements
sP [|4; 2|]     -> Min element = [|7; 11; 13|] Max element = [|997807; 997811; 997813|] of 1444 elements
sP [|6; 4; 2|]  -> Min element = [|31; 37; 41; 43|] Max element = [|997141; 997147; 997151; 997153|] of 306 elements

Factor

Works with: Factor version 0.99

<lang factor>USING: formatting fry grouping kernel math math.primes math.statistics sequences ; IN: rosetta-code.successive-prime-differences

seq-diff ( seq diffs -- seq' quot )

   dup [ length 1 + <clumps> ] dip '[ differences _ sequence= ]
   ; inline

show ( seq diffs -- )

   [ "...for differences %u:\n" printf ] keep seq-diff
   [ find nip { } like ]
   [ find-last nip { } like ]
   [ count ] 2tri
   "First group: %u\nLast group: %u\nCount: %d\n\n" printf ;

successive-prime-differences ( -- )

   "Groups of successive primes up to one million...\n" printf
   1,000,000 primes-upto {
       { 2 }
       { 1 }
       { 2 2 }
       { 2 4 }
       { 4 2 }
       { 6 4 2 }
   } [ show ] with each ;

MAIN: successive-prime-differences</lang>

Output:

Groups of successive primes up to one million...
...for differences { 2 }:
First group: { 3 5 }
Last group: { 999959 999961 }
Count: 8169

...for differences { 1 }:
First group: { 2 3 }
Last group: { 2 3 }
Count: 1

...for differences { 2 2 }:
First group: { 3 5 7 }
Last group: { 3 5 7 }
Count: 1

...for differences { 2 4 }:
First group: { 5 7 11 }
Last group: { 999431 999433 999437 }
Count: 1393

...for differences { 4 2 }:
First group: { 7 11 13 }
Last group: { 997807 997811 997813 }
Count: 1444

...for differences { 6 4 2 }:
First group: { 31 37 41 43 }
Last group: { 997141 997147 997151 997153 }
Count: 306

FreeBASIC

<lang freebasic>#include "isprime.bas"

function nextprime( n as uinteger ) as uinteger

   'finds the next prime after n
   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 spd( byval n as integer, d() as integer ) as boolean

   if not isprime(n) then return false
   for i as integer = lbound(d) to ubound(d)
       if not nextprime(n) = n + d(i) then return false
       n+=d(i)
   next i
   return true

end function

sub print_set( byval n as uinteger, d() as uinteger )

   print "( ";n;" ";
   for i as integer = lbound(d) to ubound(d)
       print n+d(i);" ";
       n+=d(i)
   next i 
   print ")"

end sub

function count_below( max as uinteger, d() as uinteger ) as uinteger

   dim as uinteger c = 0, last = 0
   for n as uinteger = 2 to max-d(ubound(d))
       if spd(n, d()) then
           c+=1
           if c=1 then print_set( n, d() )
           last = n
       end if
   next n
   print_set(last, d())
   return c

end function

dim as integer n, c

'example 1, differences of 2 redim as uinteger d(0) d(0) = 2 print "Differences of 2 (the twin primes)" c = count_below(1000000, d()) print "Number of occurrences: ", c

'example 2, difference of 1 d(0) = 1 print print "Differences of 1" c = count_below(1000000, d()) print "Number of occurrences: ", c

'example 3, differences of 2,2 redim as uinteger d(1) d(0) = 2 : d(1) = 2 print print "Differences of 2, 2" c = count_below(1000000, d()) print "Number of occurrences: ", c

'example 4, differences of 2,4 d(1) = 4 print print "Differences of 2, 4" c = count_below(1000000, d()) print "Number of occurrences: ", c

'example 5, differences of 2,2 d(0) = 4 : d(1) = 2 print print "Differences of 4, 2" c = count_below(1000000, d()) print "Number of occurrences: ", c

'example 6, differences of 6,4,2 redim as uinteger d(2) d(0) = 6 : d(1) = 4 : d(2) = 2 print print "Differences of 6, 4, 2" c = count_below(1000000, d()) print "Number of occurrences: ", c</lang>

Output:


Differences of 2 (the twin primes)
( 3 5 )
( 999959 999961 )
Number of occurrences:       8169
Differences of 1
( 2 3 )
( 2 3 )
Number of occurrences:       1
Differences of 2, 2
( 3 5 7 )
( 3 5 7 )
Number of occurrences:       1
Differences of 2, 4
( 5 7 11 )
( 999431 999433 999437 )
Number of occurrences:       1393
Differences of 4, 2
( 7 11 13 )
( 997807 997811 997813 )
Number of occurrences:       1444
Differences of 6, 4, 2
( 31 37 41 43 )
( 997141 997147 997151 997153 )
Number of occurrences:       306

Go

<lang go>package main

import "fmt"

func sieve(limit int) []int {

   primes := []int{2}
   c := make([]bool, limit+1) // composite = true
   // no need to process even numbers > 2
   p := 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
           }
       }
   }
   for i := 3; i <= limit; i += 2 {
       if !c[i] {
           primes = append(primes, i)
       }
   }
   return primes

}

func successivePrimes(primes, diffs []int) [][]int {

   var results [][]int
   dl := len(diffs)

outer:

   for i := 0; i < len(primes)-dl; i++ {
       group := make([]int, dl+1)
       group[0] = primes[i]
       for j := i; j < i+dl; j++ {
           if primes[j+1]-primes[j] != diffs[j-i] {
               group = nil
               continue outer
           }
           group[j-i+1] = primes[j+1]
       }
       results = append(results, group)
       group = nil
   }
   return results

}

func main() {

   primes := sieve(999999)
   diffsList := [][]int{{2}, {1}, {2, 2}, {2, 4}, {4, 2}, {6, 4, 2}}
   fmt.Println("For primes less than 1,000,000:-\n")
   for _, diffs := range diffsList {
       fmt.Println("  For differences of", diffs, "->")
       sp := successivePrimes(primes, diffs)
       if len(sp) == 0 {
           fmt.Println("    No groups found")
           continue
       }
       fmt.Println("    First group   = ", sp[0])
       fmt.Println("    Last group    = ", sp[len(sp)-1])
       fmt.Println("    Number found  = ", len(sp))
       fmt.Println()
   }

}</lang>

Output:

For primes less than 1,000,000:-

  For differences of [2] ->
    First group   =  [3 5]
    Last group    =  [999959 999961]
    Number found  =  8169

  For differences of [1] ->
    First group   =  [2 3]
    Last group    =  [2 3]
    Number found  =  1

  For differences of [2 2] ->
    First group   =  [3 5 7]
    Last group    =  [3 5 7]
    Number found  =  1

  For differences of [2 4] ->
    First group   =  [5 7 11]
    Last group    =  [999431 999433 999437]
    Number found  =  1393

  For differences of [4 2] ->
    First group   =  [7 11 13]
    Last group    =  [997807 997811 997813]
    Number found  =  1444

  For differences of [6 4 2] ->
    First group   =  [31 37 41 43]
    Last group    =  [997141 997147 997151 997153]
    Number found  =  306

Haskell

Uses primes library: http://hackage.haskell.org/package/primes-0.2.1.0/docs/Data-Numbers-Primes.html

Fixed computed values

<lang haskell>{-# LANGUAGE NumericUnderscores #-} import Data.Numbers.Primes (primes)

type Result = [(String, [Int])]

oneMillionPrimes :: Integral p => [p] oneMillionPrimes = takeWhile (<1_000_000) primes

getGroups :: [Int] -> Result getGroups [] = [] getGroups ps@(n:x:y:z:xs)

 | x-n == 6 && y-x == 4 && z-y == 2 = ("(6 4 2)", [n, x, y, z])            : getGroups (tail ps)
 | x-n == 4 && y-x == 2             = ("(4 2)", [n, x, y])                 : getGroups (tail ps)
 | x-n == 2 && y-x == 4             = ("(2 4)", [n, x, y]) : ("2", [n, x]) : getGroups (tail ps)
 | x-n == 2 && y-x == 2             = ("(2 2)", [n, x, y]) : ("2", [n, x]) : getGroups (tail ps)
 | x-n == 2                         = ("2", [n, x])                        : getGroups (tail ps)
 | x-n == 1                         = ("1", [n, x])                        : getGroups (tail ps)
 | otherwise                        = getGroups (tail ps)

getGroups (x:xs) = getGroups xs

groups :: Result groups = getGroups oneMillionPrimes

showGroup :: String -> IO () showGroup group = do

 putStrLn $ "Differences of " ++ group ++ ": " ++ show (length r)
 putStrLn $ "First: " ++ show (head r) ++ "\nLast:  " ++ show (last r) ++ "\n"
 where r = foldr (\(a, b) c -> if a == group then b : c else c) [] groups

main :: IO () main = showGroup "2" >> showGroup "1" >> showGroup "(2 2)" >> showGroup "(2 4)" >> showGroup "(4 2)"

 >> showGroup "(6 4 2)"</lang>

Dynamic computed input

<lang haskell>{-# LANGUAGE NumericUnderscores #-} import Data.Numbers.Primes (primes)

-- Type alias dictionary. Key is the difference group and value is a successive prime group. type Result = [(String, [Int])]

findPrimes :: [Int] -> Int -> Result findPrimes [] _ = [] findPrimes primes diffs = loopDiffs diffs <> findPrimes (tail primes) diffs

where
 loopDiffs ds = [(show d, successive)
                 | d <- ds,
                   let successive = take (length d + 1) primes,
                   subs successive == d]
 subs = map (uncurry (-)) . init . tail . (\xs -> zip (xs <> [0]) (0 : xs))

showGroup :: Result -> String -> IO () showGroup result diffs = do

 putStrLn $ "Differences of " ++ diffs ++ ": " ++ show (length groups)
 putStrLn
   $  "First: "
   ++ firstGroup groups
   ++ "\nLast:  "
   ++ lastGroup groups
   ++ "\n"
where
 groups     = [b | (a, b) <- result, a == diffs]
 firstGroup = show . head
 lastGroup  = show . last

main :: IO () main = mapM_ (showGroup result . show) diffs

where
 (diffs, result) = groups [[2], [1], [2, 2], [2, 4], [4, 2], [6, 4, 2]]
 groups diffs = (diffs, findPrimes (takeWhile (< 1_000_000) primes) diffs)</lang>

Output:

Differences of [2]: 8169
First: [3,5]
Last:  [999959,999961]

Differences of [1]: 1
First: [2,3]
Last:  [2,3]

Differences of [2,2]: 1
First: [3,5,7]
Last:  [3,5,7]

Differences of [2,4]: 1393
First: [5,7,11]
Last:  [999431,999433,999437]

Differences of [4,2]: 1444
First: [7,11,13]
Last:  [997807,997811,997813]

Differences of [6,4,2]: 306
First: [31,37,41,43]
Last:  [997141,997147,997151,997153]

J

  primes_less_than=: i.&.:(p:inv)
  assert 2 3 5 -: primes_less_than 7
  Primes=: primes_less_than 1e6

  NB. Insert minus `-/' into the length two infixes `\'.
  NB. Passive `~' swaps the arguments producing the positive differences.
  Successive_Differences=: 2 -~/\ Primes
  assert 8169 -: +/ 2 = Successive_Differences   NB. twin prime tally

  Groups=: 2 ; 1 ; 2 2 ; 2 4 ; 4 2 ; 6 4 2

  group_index=: [: I. E.
  end_groups=: Primes {~  ({. , {:)@] +/ 0,#\@[

  Header=: <;._2 'Group;Count;First/Last groups;'

  Header ,> Groups ([ ([ ; #@] ; end_groups) group_index)&.> <Successive_Differences

┌─────┬─────┬───────────────────────────┐ │Group│Count│First/Last groups │ ├─────┼─────┼───────────────────────────┤ │2 │8169 │ 3 5 │ │ │ │999959 999961 │ ├─────┼─────┼───────────────────────────┤ │1 │1 │2 3 │ │ │ │2 3 │ ├─────┼─────┼───────────────────────────┤ │2 2 │1 │3 5 7 │ │ │ │3 5 7 │ ├─────┼─────┼───────────────────────────┤ │2 4 │1393 │ 5 7 11 │ │ │ │999431 999433 999437 │ ├─────┼─────┼───────────────────────────┤ │4 2 │1444 │ 7 11 13 │ │ │ │997807 997811 997813 │ ├─────┼─────┼───────────────────────────┤ │6 4 2│306 │ 31 37 41 43│ │ │ │997141 997147 997151 997153│ └─────┴─────┴───────────────────────────┘</lang>

Java

Translation of: Go

<lang java>import java.util.ArrayList; import java.util.Arrays; import java.util.List;

public class SuccessivePrimeDifferences {

   private static Integer[] sieve(int limit) {
       List<Integer> primes = new ArrayList<>();
       primes.add(2);
       boolean[] c = new boolean[limit + 1];// composite = true
       // no need to process even numbers > 2
       int p = 3;
       while (true) {
           int p2 = p * p;
           if (p2 > limit) {
               break;
           }
           for (int i = p2; i <= limit; i += 2 * p) {
               c[i] = true;
           }
           do {
               p += 2;
           } while (c[p]);
       }
       for (int i = 3; i <= limit; i += 2) {
           if (!c[i]) {
               primes.add(i);
           }
       }

       return primes.toArray(new Integer[0]);
   }

   private static List<List<Integer>> successivePrimes(Integer[] primes, Integer[] diffs) {
       List<List<Integer>> results = new ArrayList<>();
       int dl = diffs.length;
       outer:
       for (int i = 0; i < primes.length - dl; i++) {
           Integer[] group = new Integer[dl + 1];
           group[0] = primes[i];
           for (int j = i; j < i + dl; ++j) {
               if (primes[j + 1] - primes[j] != diffs[j - i]) {
                   continue outer;
               }
               group[j - i + 1] = primes[j + 1];
           }
           results.add(Arrays.asList(group));
       }
       return results;
   }

   public static void main(String[] args) {
       Integer[] primes = sieve(999999);
       Integer[][] diffsList = {{2}, {1}, {2, 2}, {2, 4}, {4, 2}, {6, 4, 2}};
       System.out.println("For primes less than 1,000,000:-\n");
       for (Integer[] diffs : diffsList) {
           System.out.printf("  For differences of %s ->\n", Arrays.toString(diffs));
           List<List<Integer>> sp = successivePrimes(primes, diffs);
           if (sp.isEmpty()) {
               System.out.println("    No groups found");
               continue;
           }
           System.out.printf("    First group   = %s\n", Arrays.toString(sp.get(0).toArray(new Integer[0])));
           System.out.printf("    Last group    = %s\n", Arrays.toString(sp.get(sp.size() - 1).toArray(new Integer[0])));
           System.out.printf("    Number found  = %d\n", sp.size());
           System.out.println();
       }
   }

}</lang>

Output:

For primes less than 1,000,000:-

  For differences of [2] ->
    First group   = [3, 5]
    Last group    = [999959, 999961]
    Number found  = 8169

  For differences of [1] ->
    First group   = [2, 3]
    Last group    = [2, 3]
    Number found  = 1

  For differences of [2, 2] ->
    First group   = [3, 5, 7]
    Last group    = [3, 5, 7]
    Number found  = 1

  For differences of [2, 4] ->
    First group   = [5, 7, 11]
    Last group    = [999431, 999433, 999437]
    Number found  = 1393

  For differences of [4, 2] ->
    First group   = [7, 11, 13]
    Last group    = [997807, 997811, 997813]
    Number found  = 1444

  For differences of [6, 4, 2] ->
    First group   = [31, 37, 41, 43]
    Last group    = [997141, 997147, 997151, 997153]
    Number found  = 306

jq

Works with: jq

Works with gojq, the Go implementation of jq

For a suitable implementation of `is_prime` as used here, see e.g. # Erdős-primes#jq. <lang jq># Emit a stream of consecutive primes.

The stream is unbounded if . is null or infinite,
otherwise it continues up to but excluding `.`.

def primes:

 (if . == null then infinite else . end) as $n
 | 2, (range(3; $n; 2) | select(is_prime));

s is a stream
$deltas is an array
Output: a stream of arrays, each corresponding to a selection of consecutive
items from s satisfying the differences requirement.

def filter_differences(s; $deltas):

 def diffs_equal: # i.e. equal to $deltas
   . as $in
   | all( range(1;length);
          ($in[.] - $in[.-1]) == $deltas[. - 1]);

 ($deltas|length + 1) as $n
 | foreach s as $x ( {};
     .emit = null
     | .tuple += [$x]
     | .tuple |= .[-$n:]
     | if (.tuple|length) == $n
       then if (.tuple|diffs_equal) then .emit = .tuple
            else .

end else . end;

     select(.emit).emit );

def report_first_last_count(s):

 null | {first,last,count}
 | reduce s as $x (.;
     if .first == null then .first = $x else . end
     | .count = .count + 1
     | .last = $x ) ;

</lang> The Tasks <lang jq>[pow(10;6) | primes] as $p1e6 | ([2], [1], [2,2], [2,4], [4,2], [6,4,2]) as $d | ("\nFor deltas = \($d):", report_first_last_count(filter_differences($p1e6[]; $d ) ) )</lang>

Output:

For deltas = [2]:
{"first":[3,5],"last":[999959,999961],"count":8169}

For deltas = [1]:
{"first":[2,3],"last":[2,3],"count":1}

For deltas = [2,2]:
{"first":[3,5,7],"last":[3,5,7],"count":1}

For deltas = [2,4]:
{"first":[5,7,11],"last":[999431,999433,999437],"count":1393}

For deltas = [4,2]:
{"first":[7,11,13],"last":[997807,997811,997813],"count":1444}

For deltas = [6,4,2]:
{"first":[31,37,41,43],"last":[997141,997147,997151,997153],"count":306}

Julia

<lang julia>using Primes

function filterdifferences(deltas, N)

   allprimes = primes(N)
   differences = map(i -> allprimes[i + 1] - allprimes[i], 1:length(allprimes) - 1)
   println("Diff Sequence   Count         First           Last")
   for delt in deltas
       ret = trues(length(allprimes) - length(delt))
       for j in 1:length(ret)
           for (i, d) in enumerate(delt)
               if differences[j - 1 + i] != d
                   ret[j] = false
                   break
               end
           end
       end
       count, p1, pn, n = sum(ret), findfirst(ret), findlast(ret), length(delt)
       println(rpad(string(delt), 16), lpad(count, 4),
           lpad(string(allprimes[p1:p1 + n]), 18), "...", rpad(allprimes[pn:pn+n], 15))
   end

end

filterdifferences([[2], [1], [2, 2], [2, 4], [4, 2], [6, 4, 2]], 1000000)

</lang>

Output:

Diff Sequence   Count         First           Last
[2]             8169            [3, 5]...[999959, 999961]
[1]                1            [2, 3]...[2, 3]
[2, 2]             1         [3, 5, 7]...[3, 5, 7]
[2, 4]          1393        [5, 7, 11]...[999431, 999433, 999437]
[4, 2]          1444       [7, 11, 13]...[997807, 997811, 997813]
[6, 4, 2]        306  [31, 37, 41, 43]...[997141, 997147, 997151, 997153]

Lua

This task uses primegen from: Extensible_prime_generator#Lua <lang lua>function findspds(primelist, diffs)

 local results = {}
 for i = 1, #primelist-#diffs do
   result = {primelist[i]}
   for j = 1, #diffs do
     if primelist[i+j] - primelist[i+j-1] == diffs[j] then
       result[j+1] = primelist[i+j]
     else
       result = nil
       break
     end
   end
   results[#results+1] = result
 end
 return results

end

primegen:generate(nil, 1000000) for _,diffs in ipairs{{2}, {1}, {2,2}, {2,4}, {4,2}, {6,4,2}} do

 spdlist = findspds(primegen.primelist, diffs)
 print("DIFFS: ["..table.concat(diffs," ").."]")
 print("COUNT: "..#spdlist)
 print("FIRST: ["..table.concat(spdlist[1]," ").."]")
 print("LAST : ["..table.concat(spdlist[#spdlist]," ").."]")
 print()

end</lang>

Output:

DIFFS: [2]
COUNT: 8169
FIRST: [3 5]
LAST : [999959 999961]

DIFFS: [1]
COUNT: 1
FIRST: [2 3]
LAST : [2 3]

DIFFS: [2 2]
COUNT: 1
FIRST: [3 5 7]
LAST : [3 5 7]

DIFFS: [2 4]
COUNT: 1393
FIRST: [5 7 11]
LAST : [999431 999433 999437]

DIFFS: [4 2]
COUNT: 1444
FIRST: [7 11 13]
LAST : [997807 997811 997813]

DIFFS: [6 4 2]
COUNT: 306
FIRST: [31 37 41 43]
LAST : [997141 997147 997151 997153]

Kotlin

Translation of: Java

<lang scala>private fun sieve(limit: Int): Array<Int> {

   val primes = mutableListOf<Int>()
   primes.add(2)
   val c = BooleanArray(limit + 1) // composite = true
   // no need to process even numbers > 2
   var p = 3
   while (true) {
       val p2 = p * p
       if (p2 > limit) {
           break
       }
       var i = p2
       while (i <= limit) {
           c[i] = true
           i += 2 * p
       }
       do {
           p += 2
       } while (c[p])
   }
   var i = 3
   while (i <= limit) {
       if (!c[i]) {
           primes.add(i)
       }
       i += 2
   }
   return primes.toTypedArray()

}

private fun successivePrimes(primes: Array<Int>, diffs: Array<Int>): List<List<Int>> {

   val results = mutableListOf<List<Int>>()
   val dl = diffs.size
   outer@ for (i in 0 until primes.size - dl) {
       val group = IntArray(dl + 1)
       group[0] = primes[i]
       for (j in i until i + dl) {
           if (primes[j + 1] - primes[j] != diffs[j - i]) {
               continue@outer
           }
           group[j - i + 1] = primes[j + 1]
       }
       results.add(group.toList())
   }
   return results

}

fun main() {

   val primes = sieve(999999)
   val diffsList = arrayOf(
       arrayOf(2),
       arrayOf(1),
       arrayOf(2, 2),
       arrayOf(2, 4),
       arrayOf(4, 2),
       arrayOf(6, 4, 2)
   )
   println("For primes less than 1,000,000:-\n")
   for (diffs in diffsList) {
       println("  For differences of ${diffs.contentToString()} ->")
       val sp = successivePrimes(primes, diffs)
       if (sp.isEmpty()) {
           println("    No groups found")
           continue
       }
       println("    First group   = ${sp[0].toTypedArray().contentToString()}")
       println("    Last group    = ${sp[sp.size - 1].toTypedArray().contentToString()}")
       println("    Number found  = ${sp.size}")
       println()
   }

}</lang>

Output:

For primes less than 1,000,000:-

  For differences of [2] ->
    First group   = [3, 5]
    Last group    = [999959, 999961]
    Number found  = 8169

  For differences of [1] ->
    First group   = [2, 3]
    Last group    = [2, 3]
    Number found  = 1

  For differences of [2, 2] ->
    First group   = [3, 5, 7]
    Last group    = [3, 5, 7]
    Number found  = 1

  For differences of [2, 4] ->
    First group   = [5, 7, 11]
    Last group    = [999431, 999433, 999437]
    Number found  = 1393

  For differences of [4, 2] ->
    First group   = [7, 11, 13]
    Last group    = [997807, 997811, 997813]
    Number found  = 1444

  For differences of [6, 4, 2] ->
    First group   = [31, 37, 41, 43]
    Last group    = [997141, 997147, 997151, 997153]
    Number found  = 306

Mathematica/Wolfram Language

<lang Mathematica>ClearAll[Primediffs] p = Prime[Range[PrimePi[10^6]]]; Primediffs[seq_] := {First[#], Last[#], Length[#]} &[p#1 ;; #2 + 1 & @@@ SequencePosition[Differences[p], seq]] Primediffs[{2}] Primediffs[{1}] Primediffs[{2, 2}] Primediffs[{2, 4}] Primediffs[{4, 2}] Primediffs[{6, 4, 2}]</lang>

Output:

{{3,5},{999959,999961},8169}
{{2,3},{2,3},1}
{{3,5,7},{3,5,7},1}
{{5,7,11},{999431,999433,999437},1393}
{{7,11,13},{997807,997811,997813},1444}
{{31,37,41,43},{997141,997147,997151,997153},306}

Nim

<lang Nim>import math, strutils

const N = 1_000_000

var comp: array[2..(N - 1), bool] # True is composite, so default is prime. for n in 2..<N:

 if not comp[n]:
   for k in countup(n * n, N - 1, n):
     comp[k] = true

var primes = @[2] for n in countup(3, N - 1, 2):

 if not comp[n]:
   primes.add n

iterator groups(primes: seq[int]; diffs: varargs[int]): seq[int] =

 ## Yield groups of successive primes with given differences.
 var cumdiffs = cumsummed(diffs)   # Compute differences from first prime of group.
 let groupSize = diffs.len + 1
 for i in 0..(primes.len - groupSize):
   let p = primes[i]
   var group = @[p]
   for k, diff in cumdiffs:
     if primes[i + k + 1] != p + diff: break
     group.add p + diff
   if group.len == groupSize:
     yield group

proc findGroups(primes: seq[int]; diffs: varargs[int]) =

 ## In the given list of primes and for the given differences,
 ## find the first group, the last group and the count of groups.
 var
   first, last: seq[int]
   count = 0
 for group in primes.groups(diffs):
   if first.len == 0: first = group
   last = group
   inc count
 echo "Differences: ", diffs.join(", ")
 echo "– first: ($#)" % first.join(", ")
 echo "– last:  ($#)" % last.join(", ")
 echo "– count: ", count
 echo()

primes.findGroups(2) primes.findGroups(1) primes.findGroups(2, 2) primes.findGroups(2, 4) primes.findGroups(4, 2) primes.findGroups(6, 4, 2)</lang>

Output:

Differences: 2
– first: (3, 5)
– last:  (999959, 999961)
– count: 8169

Differences: 1
– first: (2, 3)
– last:  (2, 3)
– count: 1

Differences: 2, 2
– first: (3, 5, 7)
– last:  (3, 5, 7)
– count: 1

Differences: 2, 4
– first: (5, 7, 11)
– last:  (999431, 999433, 999437)
– count: 1393

Differences: 4, 2
– first: (7, 11, 13)
– last:  (997807, 997811, 997813)
– count: 1444

Differences: 6, 4, 2
– first: (31, 37, 41, 43)
– last:  (997141, 997147, 997151, 997153)
– count: 306

Perl

Library: ntheory

<lang perl>use strict; use warnings; use List::EachCons; use Array::Compare; use ntheory 'primes';

my $limit = 1E6; my @primes = (2, @{ primes($limit) }); my @intervals = map { $primes[$_] - $primes[$_-1] } 1..$#primes;

print "Groups of successive primes <= $limit\n";

my $c = Array::Compare->new; for my $diffs ([2], [1], [2,2], [2,4], [4,2], [6,4,2]) {

   my $n = -1;
   my @offsets = grep {$_} each_cons @$diffs, @intervals, sub { $n++; $n if $c->compare(\@_, \@$diffs) };
   printf "%10s has %5d sets: %15s … %s\n", 
      '(' . join(' ',@$diffs) . ')',
       scalar @offsets,
       join(' ', @primes[$offsets[ 0]..($offsets[ 0]+@$diffs)]),
       join(' ', @primes[$offsets[-1]..($offsets[-1]+@$diffs)]);

}</lang>

Output:

       (2) has  8169 sets:             3 5 … 999959 999961
       (1) has     1 sets:             2 3 … 2 3
     (2 2) has     1 sets:           3 5 7 … 3 5 7
     (2 4) has  1393 sets:          5 7 11 … 999431 999433 999437
     (4 2) has  1444 sets:         7 11 13 … 997807 997811 997813
   (6 4 2) has   306 sets:     31 37 41 43 … 997141 997147 997151 997153

Phix

with javascript_semantics
constant primes = get_primes_le(1_000_000)
procedure test(sequence differences)
    sequence res = {}
    integer ld = length(differences)
    for i=1 to length(primes)-ld do
        integer pi = primes[i]
        for j=1 to ld do
            pi += differences[j]
            if pi!=primes[i+j] then
                pi = 0
                exit
            end if
        end for
        if pi!=0 then
            res = append(res,primes[i..i+ld])
        end if
    end for
    res = {differences,length(res),res[1],res[$]}
    printf(1,"%8V : %8d %14V...%V\n",res)
end procedure
 
printf(1,"Differences   Count          First   Last\n")
papply({{2},{1},{2,2},{2,4},{4,2},{6,4,2}},test)

Output:

Differences   Count          First   Last
     {2} :     8169          {3,5}...{999959,999961}
     {1} :        1          {2,3}...{2,3}
   {2,2} :        1        {3,5,7}...{3,5,7}
   {2,4} :     1393       {5,7,11}...{999431,999433,999437}
   {4,2} :     1444      {7,11,13}...{997807,997811,997813}
 {6,4,2} :      306  {31,37,41,43}...{997141,997147,997151,997153}

Python

Uses the Sympy library.

<lang python># https://docs.sympy.org/latest/index.html from sympy import Sieve

def nsuccprimes(count, mx):

   "return tuple of <count> successive primes <= mx (generator)"
   sieve = Sieve()
   sieve.extend(mx)
   primes = sieve._list
   return zip(*(primes[n:] for n in range(count)))

def check_value_diffs(diffs, values):

   "Differences between successive values given by successive items in diffs?"
   return all(v[1] - v[0] == d 
              for d, v in zip(diffs, zip(values, values[1:])))

def successive_primes(offsets=(2, ), primes_max=1_000_000):

   return (sp for sp in nsuccprimes(len(offsets) + 1, primes_max) 
           if check_value_diffs(offsets, sp))

if __name__ == '__main__':

   for offsets, mx in [((2,),      1_000_000), 
                       ((1,),      1_000_000),
                       ((2, 2),    1_000_000),
                       ((2, 4),    1_000_000),
                       ((4, 2),    1_000_000),
                       ((6, 4, 2), 1_000_000),
                      ]:
       print(f"## SETS OF {len(offsets)+1} SUCCESSIVE PRIMES <={mx:_} WITH "
             f"SUCCESSIVE DIFFERENCES OF {str(list(offsets))[1:-1]}")
       for count, last in enumerate(successive_primes(offsets, mx), 1):
           if count == 1:
               first = last
       print("  First group:", str(first)[1:-1])
       print("   Last group:", str(last)[1:-1])
       print("        Count:", count)</lang>

Output:

## SETS OF 2 SUCCESSIVE PRIMES <=1_000_000 WITH SUCCESSIVE DIFFERENCES OF 2
  First group: 3, 5
   Last group: 999959, 999961
        Count: 8169
## SETS OF 2 SUCCESSIVE PRIMES <=1_000_000 WITH SUCCESSIVE DIFFERENCES OF 1
  First group: 2, 3
   Last group: 2, 3
        Count: 1
## SETS OF 3 SUCCESSIVE PRIMES <=1_000_000 WITH SUCCESSIVE DIFFERENCES OF 2, 2
  First group: 3, 5, 7
   Last group: 3, 5, 7
        Count: 1
## SETS OF 3 SUCCESSIVE PRIMES <=1_000_000 WITH SUCCESSIVE DIFFERENCES OF 2, 4
  First group: 5, 7, 11
   Last group: 999431, 999433, 999437
        Count: 1393
## SETS OF 3 SUCCESSIVE PRIMES <=1_000_000 WITH SUCCESSIVE DIFFERENCES OF 4, 2
  First group: 7, 11, 13
   Last group: 997807, 997811, 997813
        Count: 1444
## SETS OF 4 SUCCESSIVE PRIMES <=1_000_000 WITH SUCCESSIVE DIFFERENCES OF 6, 4, 2
  First group: 31, 37, 41, 43
   Last group: 997141, 997147, 997151, 997153
        Count: 306

Raku

(formerly Perl 6)

Categorized by Successive

Works with: Rakudo version 2019.03

Essentially the code from the Sexy primes task with minor tweaks.

<lang perl6>use Math::Primesieve; my $sieve = Math::Primesieve.new;

my $max = 1_000_000; my @primes = $sieve.primes($max); my $filter = @primes.Set; my $primes = @primes.categorize: &successive;

sub successive ($i) {

   gather {
       take '2' if $filter{$i + 2};
       take '1' if $filter{$i + 1};
       take '2_2' if all($filter{$i «+« (2,4)});
       take '2_4' if all($filter{$i «+« (2,6)});
       take '4_2' if all($filter{$i «+« (4,6)});
       take '6_4_2' if all($filter{$i «+« (6,10,12)}) and
           none($filter{$i «+« (2,4,8)});
   }

}

sub comma { $^i.flip.comb(3).join(',').flip }

for (2,), (1,), (2,2), (2,4), (4,2), (6,4,2) -> $succ {

   say "## Sets of {1+$succ} successive primes <= { comma $max } with " ~
       "successive differences of { $succ.join: ', ' }";
   my $i = $succ.join: '_';
   for 'First', 0, ' Last', * - 1 -> $where, $ind {
       say "$where group: ", join ', ', [\+] flat $primes{$i}[$ind], |$succ
   }
   say '      Count: ', +$primes{$i}, "\n";

}</lang>

Output:

## Sets of 2 successive primes <= 1,000,000 with successive differences of 2
First group: 3, 5
 Last group: 999959, 999961
      Count: 8169

## Sets of 2 successive primes <= 1,000,000 with successive differences of 1
First group: 2, 3
 Last group: 2, 3
      Count: 1

## Sets of 3 successive primes <= 1,000,000 with successive differences of 2, 2
First group: 3, 5, 7
 Last group: 3, 5, 7
      Count: 1

## Sets of 3 successive primes <= 1,000,000 with successive differences of 2, 4
First group: 5, 7, 11
 Last group: 999431, 999433, 999437
      Count: 1393

## Sets of 3 successive primes <= 1,000,000 with successive differences of 4, 2
First group: 7, 11, 13
 Last group: 997807, 997811, 997813
      Count: 1444

## Sets of 4 successive primes <= 1,000,000 with successive differences of 6, 4, 2
First group: 31, 37, 41, 43
 Last group: 997141, 997147, 997151, 997153
      Count: 306

Precomputed Differences

Works with: Rakudo version 2019.03

<lang perl6>use Math::Primesieve;

constant $max = 1_000_000; constant @primes = Math::Primesieve.primes($max); constant @diffs = @primes.skip Z- @primes;

say "Given all ordered primes <= $max, sets with successive differences of:"; for (2,), (1,), (2,2), (2,4), (4,2), (6,4,2) -> @succ {

   my $size = @succ.elems;

   my @group_start_offsets = @diffs.rotor( $size => 1-$size )
                                   .grep(:k, { $_ eqv @succ });

   my ($first, $last) = @group_start_offsets[0, *-1]
                        .map: { @primes.skip($_).head($size + 1) };

   say sprintf '%10s has %5d sets: %15s … %s',
       @succ.gist, @group_start_offsets.elems, $first, $last;

}</lang>

Output:

Given all ordered primes <= 1000000, sets with successive differences of:
       (2) has  8169 sets:             3 5 … 999959 999961
       (1) has     1 sets:             2 3 … 2 3
     (2 2) has     1 sets:           3 5 7 … 3 5 7
     (2 4) has  1393 sets:          5 7 11 … 999431 999433 999437
     (4 2) has  1444 sets:         7 11 13 … 997807 997811 997813
   (6 4 2) has   306 sets:     31 37 41 43 … 997141 997147 997151 997153

REXX

<lang rexx>/*REXX program finds and displays primes with successive differences (up to a limit).*/ parse arg H . 1 . difs /*allow the highest number be specified*/ if H== | H=="," then H= 1000000 /*Not specified? Then use the default.*/ if difs= then difs= 2 1 2.2 2.4 4.2 6.4.2 /* " " " " " " */ call genP H

       do j=1  for words(difs)                             /*traipse through the lists.*/
       dif= translate( word(difs, j),,.);  dw= words(dif)  /*obtain true differences.  */
           do i=1  for dw;  dif.i= word(dif, i)            /*build an array of diffs.  */
           end   /*i*/                                     /* [↑]  for optimization.   */
       say center('primes with differences of:'  dif,  50, '─')        /*display title.*/
       p= 1;                        c= 0;        grp=      /*init. prime#,  count, grp.*/
            do a=1;  p= nextP(p+1);  if p==0  then leave   /*find the next  DIF  primes*/
            aa= p;   !.=                                   /*AA: nextP;  the group #'s.*/
            !.1= p                                         /*assign 1st prime in group.*/
                   do g=2  for dw                          /*get the rest of the group.*/
                   aa= nextP(aa+1); if aa==0  then leave a /*obtain the next prime.    */
                   !.g= aa;         _= g-1                 /*prepare to add difference.*/
                   if !._ + dif._\==!.g  then iterate a    /*determine if fits criteria*/
                   end   /*g*/
            c= c+1                                         /*bump count of # of groups.*/
            grp= !.1;       do b=2  for dw;  grp= grp !.b  /*build a list of primes.   */
                            end   /*b*/
            if c==1  then say '     first group: '   grp   /*display the first group.  */
            end   /*a*/
                                                           /* [↓]  test if group found.*/
       if grp==   then say "         (none)"             /*display the  last group.  */
                    else say '      last group: '   grp    /*   "     "     "    "     */
                         say '           count: '   c      /*   "     "  group count.  */
       say
       end   /*j*/

exit /*stick a fork in it, we're all done. */ /*──────────────────────────────────────────────────────────────────────────────────────*/ nextP: do nxt=arg(1) to H; if @.nxt==. then return nxt; end /*nxt*/; return 0 /*──────────────────────────────────────────────────────────────────────────────────────*/ genP: procedure expose @.; parse arg N; != 0; @.=.; @.1= /*initialize the array.*/

         do e=4  by 2  for (N-1)%2;  @.e=;  end /*treat the even integers > 2  special.*/
                                                /*all primes are indicated with a  "." */
       do j=1  by 2  for (N-1)%2                /*use odd integers up to  N  inclusive.*/
       if @.j==.  then do;  if !  then iterate  /*Prime?   Should skip the top part ?  */
                            jj= j * j           /*compute the square of  J.        ___ */
                            if jj>N  then != 1  /*indicate skip top part  if  j > √ N  */
                              do m=jj  to N  by j+j;  @.m=;  end       /*odd multiples.*/
                       end                      /* [↑]  strike odd multiples  ¬ prime. */
       end   /*j*/;               return</lang>

output when using the default inputs:

──────────primes with differences of: 2───────────
     first group:  3 5
      last group:  999959 999961
           count:  8169

──────────primes with differences of: 1───────────
     first group:  2 3
      last group:  2 3
           count:  1

─────────primes with differences of: 2 2──────────
     first group:  3 5 7
      last group:  3 5 7
           count:  1

─────────primes with differences of: 2 4──────────
     first group:  5 7 11
      last group:  999431 999433 999437
           count:  1393

─────────primes with differences of: 4 2──────────
     first group:  7 11 13
      last group:  997807 997811 997813
           count:  1444

────────primes with differences of: 6 4 2─────────
     first group:  31 37 41 43
      last group:  997141 997147 997151 997153
           count:  306

Ring

<lang ring> load "stdlib.ring" see "working..." + nl + nl see "For primes less than 1,000,000:" + nl + nl num = 0 limit = 1000000 Primes = [] SuccPrimes = [] Sp = [[2],[1],[2,2],[2,4],[4,2],[6,4,2]]

for n = 1 to limit

   if isprime(n)
      add(Primes,n)
   ok

   num = 0
   for m = 1 to len(Primes)-len(Sp[n])
       flag = 0
       SuccPrimes = []
       for p = 1 to len(Sp[n])
           if (Primes[m+p]-Primes[m+p-1] = Sp[n][p]) 
              flag++
              add(SuccPrimes,Primes[m+p])
              add(SuccPrimes,Primes[m+p-1])
           else
              exit
           ok
       next

       SuccPrimes = sort(SuccPrimes)
       for x = len(SuccPrimes) to 2 step -1
           if SuccPrimes[x] = SuccPrimes[x-1]
              del(SuccPrimes,x)
           ok
       next

       if len(SuccPrimes) = len(Sp[n])+1
          num++
          LastSuccPrimes = SuccPrimes
          if num = 1
             see "For differences of " 
             showArray(Sp[n])
             see " ->" + nl
             see " First group = "
             showArray(SuccPrimes)
             see nl
          ok
       ok
   next
   see " Last group = "
   showArray(LastSuccPrimes)
   see nl
   see " Number found = " + num + nl + nl
next

see "done..." + nl

func showArray(array)

    txt = ""
    see "["
    for n = 1 to len(array)
        txt = txt + array[n] + ","
    next
    txt = left(txt,len(txt)-1)
    txt = txt + "]"
    see txt

</lang>

Output:

working...

For primes less than 1,000,000:

For differences of [2] ->
 First group = [3,5]
 Last group = [999959,999961]
 Number found = 8169

For differences of [1] ->
 First group = [2,3]
 Last group = [2,3]
 Number found = 1

For differences of [2,2] ->
 First group = [3,5,7]
 Last group = [3,5,7]
 Number found = 1

For differences of [2,4] ->
 First group = [5,7,11]
 Last group = [999431,999433,999437]
 Number found = 1393

For differences of [4,2] ->
 First group = [7,11,13]
 Last group = [997807,997811,997813]
 Number found = 1444

For differences of [6,4,2] ->
 First group = [31,37,41,43]
 Last group = [997141,997147,997151,997153]
 Number found = 306

done...

Ruby

<lang ruby>require 'prime' PRIMES = Prime.each(1_000_000).to_a difs = [[2], [1], [2,2], [2,4], [4,2], [6,4,2]]

difs.each do |ar|

 res = PRIMES.each_cons(ar.size+1).select do |slice|
   slice.each_cons(2).zip(ar).all? {|(a,b), c| a+c == b}
 end
 puts "#{ar} has #{res.size} sets. #{res.first}...#{res.last}"

end </lang>

Output:

[2] has 8169 sets. [3, 5]...[999959, 999961]
[1] has 1 sets. [2, 3]...[2, 3]
[2, 2] has 1 sets. [3, 5, 7]...[3, 5, 7]
[2, 4] has 1393 sets. [5, 7, 11]...[999431, 999433, 999437]
[4, 2] has 1444 sets. [7, 11, 13]...[997807, 997811, 997813]
[6, 4, 2] has 306 sets. [31, 37, 41, 43]...[997141, 997147, 997151, 997153]

Scala

<lang scala>object SuccessivePrimeDiffs {

 def main(args: Array[String]): Unit = {
   val d2 = primesByDiffs(2)(1000000)
   val d1 = primesByDiffs(1)(1000000)
   val d22 = primesByDiffs(2, 2)(1000000)
   val d24 = primesByDiffs(2, 4)(1000000)
   val d42 = primesByDiffs(4, 2)(1000000)
   val d642 = primesByDiffs(6, 4, 2)(1000000)
   
   if(true) println(
     s"""|Diffs: (First), (Last), Count
         |2:     (${d2.head.mkString(", ")}), (${d2.last.mkString(", ")}), ${d2.size}
         |1:     (${d1.head.mkString(", ")}), (${d1.last.mkString(", ")}), ${d1.size}
         |2-2:   (${d22.head.mkString(", ")}), (${d22.last.mkString(", ")}), ${d22.size}
         |2-4:   (${d24.head.mkString(", ")}), (${d24.last.mkString(", ")}), ${d24.size}
         |4-2:   (${d42.head.mkString(", ")}), (${d42.last.mkString(", ")}), ${d42.size}
         |6-4-2: (${d642.head.mkString(", ")}), (${d642.last.mkString(", ")}), ${d642.size}
         |""".stripMargin)
 }
 
 def primesByDiffs(diffs: Int*)(max: Int): LazyList[Vector[Int]] = {
   primesSliding(diffs.size + 1)
     .takeWhile(_.last <= max)
     .filter{vec => diffs.zip(vec.init).map{case (a, b) => a + b} == vec.tail}
     .to(LazyList)
 }
 
 def primesSliding(len: Int): Iterator[Vector[Int]] = primes.sliding(len).map(_.toVector)
 def primes: LazyList[Int] = 2 #:: LazyList.from(3, 2).filter(n => !Iterator.range(3, math.sqrt(n).toInt + 1, 2).exists(n%_ == 0))

}</lang>

Output:

Diffs: (First), (Last), Count
2:     (3, 5), (999959, 999961), 8169
1:     (2, 3), (2, 3), 1
2-2:   (3, 5, 7), (3, 5, 7), 1
2-4:   (5, 7, 11), (999431, 999433, 999437), 1393
4-2:   (7, 11, 13), (997807, 997811, 997813), 1444
6-4-2: (31, 37, 41, 43), (997141, 997147, 997151, 997153), 306

Sidef

<lang ruby>var limit = 1e6 var primes = limit.primes

say "Groups of successive primes <= #{limit.commify}:"

for diffs in [[2], [1], [2,2], [2,4], [4,2], [6,4,2]] {

   var groups = []
   primes.each_cons(diffs.len+1, {|*group|
       if (group.map_cons(2, {|a,b| b-a}) == diffs) {
           groups << group
       }
   })

   say ("...for differences #{diffs}, there are #{groups.len} groups, where ",
        "the first group = #{groups.first} and the last group = #{groups.last}")

}</lang>

Output:

Groups of successive primes <= 1,000,000:
...for differences [2], there are 8169 groups, where the first group = [3, 5] and the last group = [999959, 999961]
...for differences [1], there are 1 groups, where the first group = [2, 3] and the last group = [2, 3]
...for differences [2, 2], there are 1 groups, where the first group = [3, 5, 7] and the last group = [3, 5, 7]
...for differences [2, 4], there are 1393 groups, where the first group = [5, 7, 11] and the last group = [999431, 999433, 999437]
...for differences [4, 2], there are 1444 groups, where the first group = [7, 11, 13] and the last group = [997807, 997811, 997813]
...for differences [6, 4, 2], there are 306 groups, where the first group = [31, 37, 41, 43] and the last group = [997141, 997147, 997151, 997153]

Visual Basic .NET

Translation of: Java

<lang vbnet>Imports System.Text

Module Module1

   Function Sieve(limit As Integer) As Integer()
       Dim primes As New List(Of Integer) From {2}
       Dim c(limit + 1) As Boolean REM composite = true
       REM no need to process even numbers > 2
       Dim p = 3
       While True
           Dim p2 = p * p
           If p2 > limit Then
               Exit While
           End If
           For i = p2 To limit Step 2 * p
               c(i) = True
           Next
           Do
               p += 2
           Loop While c(p)
       End While
       For i = 3 To limit Step 2
           If Not c(i) Then
               primes.Add(i)
           End If
       Next
       Return primes.ToArray
   End Function

   Function SuccessivePrimes(primes() As Integer, diffs() As Integer) As List(Of List(Of Integer))
       Dim results As New List(Of List(Of Integer))
       Dim dl = diffs.Length
       Dim i = 0
       While i < primes.Length - dl
           Dim group(dl) As Integer
           group(0) = primes(i)

           Dim j = i
           While j < i + dl
               If primes(j + 1) - primes(j) <> diffs(j - i) Then
                   GoTo outer REM continue the outermost loop
               End If
               group(j - i + 1) = primes(j + 1)

               j += 1
           End While
           results.Add(group.ToList)

outer:

           i += 1
       End While
       Return results
   End Function

   Function CollectionToString(Of T)(c As IEnumerable(Of T)) As String
       Dim builder As New StringBuilder
       builder.Append("[")

       Dim it = c.GetEnumerator()
       If it.MoveNext() Then
           builder.Append(it.Current)
       End If
       While it.MoveNext()
           builder.Append(", ")
           builder.Append(it.Current)
       End While

       builder.Append("]")
       Return builder.ToString
   End Function

   Sub Main()
       Dim primes = Sieve(999999)
       Dim diffsList = {({2}), ({1}), ({2, 2}), ({2, 4}), ({4, 2}), ({6, 4, 2})}
       Console.WriteLine("For primes less than 1,000,000:-")
       Console.WriteLine()
       For Each diffs In diffsList
           Console.WriteLine("  For differences of {0} ->", CollectionToString(diffs))
           Dim sp = SuccessivePrimes(primes, diffs)
           If sp.Count = 0 Then
               Console.WriteLine("    No groups found")
               Continue For
           End If
           Console.WriteLine("    First group   = {0}", CollectionToString(sp(0)))
           Console.WriteLine("    Last group    = {0}", CollectionToString(sp(sp.Count - 1)))
           Console.WriteLine("    Number found  = {0}", sp.Count)
           Console.WriteLine()
       Next
   End Sub

End Module</lang>

Output:

For primes less than 1,000,000:-

  For differences of [2] ->
    First group   = [3, 5]
    Last group    = [999959, 999961]
    Number found  = 8169

  For differences of [1] ->
    First group   = [2, 3]
    Last group    = [2, 3]
    Number found  = 1

  For differences of [2, 2] ->
    First group   = [3, 5, 7]
    Last group    = [3, 5, 7]
    Number found  = 1

  For differences of [2, 4] ->
    First group   = [5, 7, 11]
    Last group    = [999431, 999433, 999437]
    Number found  = 1393

  For differences of [4, 2] ->
    First group   = [7, 11, 13]
    Last group    = [997807, 997811, 997813]
    Number found  = 1444

  For differences of [6, 4, 2] ->
    First group   = [31, 37, 41, 43]
    Last group    = [997141, 997147, 997151, 997153]
    Number found  = 306

Wren

Translation of: Go

Library: Wren-math

<lang ecmascript>import "/math" for Int

var successivePrimes = Fn.new { |primes, diffs|

   var results = []
   var dl = diffs.count
   for (i in 0...primes.count-dl) {
       var group = List.filled(dl+1, 0)
       group[0] = primes[i]
       var outer = false
       for (j in i...i+dl) {
           var cont = false
           if (primes[j+1] - primes[j] != diffs[j-i]) {
               outer = true
               break             
           }
           group[j-i+1] = primes[j+1]
       }
       if (!outer) results.add(group)
   }
   return results

}

var primes = Int.primeSieve(999999) var diffsList = [ [2], [1], [2, 2], [2, 4], [4, 2], [6, 4, 2] ] System.print("For primes less than 1,000,000:-\n") for (diffs in diffsList) {

   System.print("  For differences of %(diffs) ->")
   var sp = successivePrimes.call(primes, diffs)
   var cont = false
   if (sp.count == 0) {
       System.print("    No groups found")
       cont = true
   }
   if (!cont) {
       System.print("    First group   =  %(sp[0])")
       System.print("    Last group    =  %(sp[-1])")
       System.print("    Number found  =  %(sp.count)\n")
   }

}</lang>

Output:

For primes less than 1,000,000:-

  For differences of [2] ->
    First group   =  [3, 5]
    Last group    =  [999959, 999961]
    Number found  =  8169

  For differences of [1] ->
    First group   =  [2, 3]
    Last group    =  [2, 3]
    Number found  =  1

  For differences of [2, 2] ->
    First group   =  [3, 5, 7]
    Last group    =  [3, 5, 7]
    Number found  =  1

  For differences of [2, 4] ->
    First group   =  [5, 7, 11]
    Last group    =  [999431, 999433, 999437]
    Number found  =  1393

  For differences of [4, 2] ->
    First group   =  [7, 11, 13]
    Last group    =  [997807, 997811, 997813]
    Number found  =  1444

  For differences of [6, 4, 2] ->
    First group   =  [31, 37, 41, 43]
    Last group    =  [997141, 997147, 997151, 997153]
    Number found  =  306

zkl

Library: GMP

GNU Multiple Precision Arithmetic Library

Using GMP ( probabilistic primes), because it is easy and fast to generate primes.

Extensible prime generator#zkl could be used instead.

Treat this as a string search problem. <lang zkl>const PRIME_LIMIT=1_000_000; var [const] BI=Import("zklBigNum"); // libGMP var [const] primeBitMap=Data(PRIME_LIMIT).fill(0x30); // one big string p:= BI(1); while(p.nextPrime()<=PRIME_LIMIT){ primeBitMap[p]="1" } // bitmap of primes

fcn primeWindows(m,deltas){ // eg (6,4,2)

  n,r := 0,List();
  ds:=deltas.len().pump(List,'wrap(n){ deltas[0,n+1].sum(0) });  // (6,10,12)
  sp:=Data(Void,"1" + "0"*deltas.sum(0));
  foreach n in (ds){ sp[n]="1" } // "1000001000101" 
  while(n=primeBitMap.find(sp,n+1)){ r.append(n) }  //  (31, 61, 271,...)
  r.apply('wrap(n){ T(n).extend(ds.apply('+(n))) }) //( (31,37,41,43), (61,67,71,73), (271,277,281,283) ...)

}</lang> <lang zkl>foreach w in (T( T(2), T(1), T(2,2), T(2,4), T(4,2), T(6,4,2) )){

  r:=primeWindows(PRIME_LIMIT,w);
  println("Successive primes (<=%,d) with deltas of %s (%,d groups):"
     .fmt(PRIME_LIMIT,w.concat(","),r.len()));
  println("   First group: %s;  Last group: %s\n"
     .fmt(r[0].concat(", "),r[-1].concat(", ")));

}</lang>

Output:

Successive primes (<=1,000,000) with deltas of 2 (8,169 groups):
   First group: 3, 5;  Last group: 999959, 999961

Successive primes (<=1,000,000) with deltas of 1 (1 groups):
   First group: 2, 3;  Last group: 2, 3

Successive primes (<=1,000,000) with deltas of 2,2 (1 groups):
   First group: 3, 5, 7;  Last group: 3, 5, 7

Successive primes (<=1,000,000) with deltas of 2,4 (1,393 groups):
   First group: 5, 7, 11;  Last group: 999431, 999433, 999437

Successive primes (<=1,000,000) with deltas of 4,2 (1,444 groups):
   First group: 7, 11, 13;  Last group: 997807, 997811, 997813

Successive primes (<=1,000,000) with deltas of 6,4,2 (306 groups):
   First group: 31, 37, 41, 43;  Last group: 997141, 997147, 997151, 997153