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
# 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 ) ;
 