Initial members of "L - consecutive prime k-tuplets with the smallest possible and constant gap (D)"

Consecutive Prime Twins
L D First initial members Pattern d n's Where Who
2 6  [ 5 + d ] + 6 · n d = 0, 2 0,1
3 12  [ 5 + d ] + 12 · n d = 0, 2 0..2
4 30  [ 11 + d ] + 30 · n d = 0, 2 0..3
5 30  [ 1702218611 + d ] + 30 · n d = 0, 2 0..4
6 210  [ 3042491 + d ] + 210 · n d = 0, 2 0..504 Mar 2025 Aliaksei Strakh
7 210  [ 66583737656789 + d ] + 210 · n d = 0, 2 0..6
8 210  [ 40947105152901707 + d ] + 210 · n d = 0, 2 0..718 Feb 2025 Aliaksei Strakh
9 1890  [ 2705965414000880284271 + d ] + 1890 · n d = 0, 2 0..804 Mar 2025 Aliaksei Strakh

Consecutive Prime Triplets
L D First initial members Pattern d n's Where Who
2 6
6		  [ 5 + d ] + 6 · n
 [ 7 + d ] + 6 · n 		d = 0, 2, 6
d = 0, 4, 6		0,1
3 30
30		  [ 97547 + d ] + 30 · n
 [ 7 + d ] + 30 · n 		d = 0, 2, 6
d = 0, 4, 6		0..2
4 210
210		  [ 38680961771 + d ] + 210 · n
 [ 2783876437 + d ] + 210 · n 		d = 0, 2, 6
d = 0, 4, 6		0..3
5 420
420		  [ 1063383716622671 + d ] + 420 · n
 [ 366943408969447 + d ] + 420 · n 		d = 0, 2, 6
d = 0, 4, 6		0..4
6 420
420		  [ 38031991346321640527 + d ] + 420 · n
 [ ? + d ] + 420 · n 		d = 0, 2, 6
d = 0, 4, 6		0..5 05 Mar 2025  Aliaksei Strakh

Consecutive Prime Quadruplets
L D First initial members Pattern d n's Where Who
2 30  [ 1006301 + d ] + 30 · n
d = 0, 2, 6, 80,1
3 90  [ 11 + d ] + 90 · n
d = 0, 2, 6, 80..2
4 420  [ 6234140990672051 + d ] + 420 · n
d = 0, 2, 6, 80..3

Consecutive Prime Quintuplets
L D First initial members Pattern d n's Where Who
2 90
90		  [ 11 + d ] + 90 · n
 [ 7 + d ] + 90 · n 		d = 0, 2, 6, 8, 12
d = 0, 4, 6, 10, 12		0,1
3 210
210		  [ 24097678829994941 + d ] + 210 · n
 [ 74426354260647787 + d ] + 210 · n 		d = 0, 2, 6, 8, 12
d = 0, 4, 6, 10, 12		0..2

Consecutive Prime Sextuplets
L D First initial members Pattern d n's Where Who
2 210  [ 5835906544537 + d ] + 210 · n
d = 0, 4, 6, 10, 12, 160,1
3 210  [ 50038627250687303646277 + d ] + 210 · n
d = 0, 4, 6, 10, 12, 160..2  Jörg Waldvogel & Peter Leikauf

Consecutive Prime Septuplets
L D First initial members Pattern d n's Where Who
2 210
210		  [ 1683059174212301+ d ] + 210 · n
 [ 482046424357019 + d ] + 210 · n 		d = 0, 2, 6, 8, 12, 18, 20
d = 0, 2, 8, 12, 14, 18, 20		0,1

Consecutive Prime Octuplets
L D First initial members Pattern d n's Where Who
2 60
420
420		  [ 10458834002271815117 + d ] + 60 · n
 [ 7387829334225124901 + d ] + 420 · n
 [ 1365135293377079663 + d ] + 420 · n 		d = 0, 2, 6, 12, 14, 20, 24, 26
d = 0, 2, 6, 8, 12, 18, 20, 26
d = 0, 6, 8, 14, 18, 20, 24, 26		0,1

Consecutive Prime Nonuplets
L D First initial members Pattern d n's Where Who
2 60  [ 54014646858393564373 + d ] + 60 · n
d = 0, 4, 6, 10, 16, 18, 24, 28, 300,1 Jörg Waldvogel & Peter Leikauf