Prime p₀= 4276877365741624871 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 34 2 100 2 70 2 4 2 82 2 46 2 (?)
a_n: 0 2 36 38 138 140 210 212 216 218 300 302 348 350
All 14 integers p_n = p₀+a_n = p_n-1+d_n are prime and there are no other primes in the gaps.
diameter: 350
Found by: Autofuzzy
db: batch=85, k=14, 34 100 70 4 82 46

SPT(16)	4276876355382553177	0 24 30 66 84 100 124 132 184 192 216 232 250 286 292 316	-1010359071694
SPT(16)	4276876554252960529	0 4 40 70 78 114 120 172 180 232 238 274 282 312 348 352	-811488664342
SPT(16)	4276876598233013473	0 34 90 108 156 168 184 186 208 210 226 238 286 304 360 394	-767508611398
STPT(12)	4276876816511697551	0 2 18 20 60 62 96 98 138 140 156 158	-549229927320
SPT(16)	4276877102922310493	0 24 26 48 54 68 74 80 84 90 96 110 116 138 140 164	-262819314378
STPT(10)	4276877231638974299	0 2 30 32 120 122 210 212 240 242	-134102650572
SPT(16)	4276877374402688309	0 20 38 60 144 150 224 258 260 294 368 374 458 480 498 518	8661063438
SPT(16)	4276877630911980959	0 2 8 12 44 50 90 132 140 182 222 228 260 264 270 272	265170356088
SPT(16)	4276877657202024827	0 12 42 54 96 110 120 140 156 176 186 200 242 254 284 296	291460399956
SPT(16)	4276877742432642637	0 12 52 66 70 96 120 156 286 322 346 372 376 390 430 442	376691017766
STPT(10)	4276877937461739227	0 2 72 74 102 104 132 134 204 206	571720114356