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
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) ...)
}