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
function findspds(primelist, diffs)
  local results = {}
  for i = 1, #primelist-#diffs do
    result = {primelist[i]}
    for j = 1, #diffs do
      if primelist[i+j] - primelist[i+j-1] == diffs[j] then
        result[j+1] = primelist[i+j]
      else
        result = nil
        break
      end
    end
    results[#results+1] = result
  end
  return results
end
 
primegen:generate(nil, 1000000)
for _,diffs in ipairs{{2}, {1}, {2,2}, {2,4}, {4,2}, {6,4,2}} do
  spdlist = findspds(primegen.primelist, diffs)
  print("DIFFS: ["..table.concat(diffs," ").."]")
  print("COUNT: "..#spdlist)
  print("FIRST: ["..table.concat(spdlist[1]," ").."]")
  print("LAST : ["..table.concat(spdlist[#spdlist]," ").."]")
  print()
end