Prime p₀= 4387550971688445161 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 28 2 64 2 28 2 40 2 40 2 16 2 (?)
a_n: 0 2 30 32 96 98 126 128 168 170 210 212 228 230
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: 230
Found by: [SG]KidDoesCrunch
db: batch=85, k=14, 28 64 28 40 40 16

SPT(16)	4387549554637387339	0 48 64 78 84 112 130 148 174 192 210 238 244 258 274 322	-1417051057822
SPT(16)	4387549986920555311	0 96 106 112 150 162 180 216 226 262 280 292 330 336 346 442	-984767889850
STPT(10)	4387550380769946491	0 2 6 8 48 50 90 92 96 98	-590918498670
STPT(10)	4387550617265639729	0 2 12 14 90 92 168 170 180 182	-354422805432
SPT(16)	4387550656771807307	0 14 26 66 182 194 216 222 224 230 252 264 380 420 432 446	-314916637854
SPT(16)	4387550765325338137	0 24 52 90 102 114 130 132 154 156 172 184 196 234 262 286	-206363107024
TPT(14)	4387551166186511417	0 2 54 56 90 92 174 176 180 182 204 206 222 224	194498066256
SPT(16)	4387551546514450123	0 84 94 118 124 138 150 160 228 238 250 264 270 294 304 388	574826004962
SPT(18)	4387551722493389531	0 6 26 98 132 158 180 236 242 246 252 308 330 356 390 462 482 488	750804944370
STPT(10)	4387552829159436647	0 2 42 44 72 74 102 104 144 146	1857470991486
SPT(16)	4387552961795780809	0 22 40 42 60 64 102 138 214 250 288 292 310 312 330 352	1990107335648