Prime p₀= 4387497287196510239 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 16 2 22 2 58 2 124 2 178 2 22 2 (?)
a_n: 0 2 18 20 42 44 102 104 228 230 408 410 432 434
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: 434
Found by: [SG]KidDoesCrunch
db: batch=85, k=14, 16 22 58 124 178 22

STPT(10)	4387495974943901819	0 2 18 20 30 32 42 44 60 62	-1312252608420
SPT(16)	4387496418377906153	0 8 36 66 80 90 116 126 158 168 194 204 218 248 276 284	-868818604086
SPT(16)	4387496487254519329	0 22 42 48 52 70 132 150 172 190 252 270 274 280 300 322	-799941990910
SPT(16)	4387496808903885389	0 32 62 114 138 144 170 182 240 252 278 284 308 360 390 422	-478292624850
SPT(16)	4387497035157166981	0 10 12 36 40 58 66 72 136 142 150 168 172 196 198 208	-252039343258
SPT(16)	4387497175980816719	0 38 42 44 72 78 128 132 170 174 224 230 258 260 264 302	-111215693520
SPT(16)	4387497439618247137	0 22 24 30 42 46 94 102 154 162 210 214 226 232 234 256	152421736898
TPT(14)	4387497489470166317	0 2 72 74 162 164 240 242 330 332 360 362 462 464	202273656078
STPT(10)	4387498511948481899	0 2 12 14 30 32 48 50 60 62	1224751971660
STPT(10)	4387498575333870209	0 2 12 14 90 92 168 170 180 182	1288137359970
STPT(10)	4387498818540290249	0 2 18 20 30 32 42 44 60 62	1531343780010