Task: Successive prime differences
C#: 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;
        }
    }
}

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];
        
        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();
        }
    }
}