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
https://pdfs.semanticscholar.org/78a1/7349819304863ae061df88dbcb26b4908f03.pdf
https://www.primepuzzles.net/puzzles/puzz_011.htm
https://matheplanet.de/matheplanet/nuke/html/viewtopic.php?topic=232720&start=0
# 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)