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
 
   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│
└─────┴─────┴───────────────────────────┘