Prime p₀= 4364178486353055659 starts a Tuple of 14 consecutive primes, which form 7 twin primes (TPT).
d_n: 0 2 40 2 4 2 28 2 40 2 10 2 34 2 (?)
a_n: 0 2 42 44 48 50 78 80 120 122 132 134 168 170
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: 170
Found by: Autofuzzy
db: batch=85, k=14, 40 4 28 40 10 34

SPT(16)	4364177179159848709	0 42 52 70 102 120 130 138 154 162 172 190 222 240 250 292	-1307193206950
SPT(16)	4364177513090887861	0 6 12 42 52 90 96 102 160 166 172 210 220 250 256 262	-973262167798
SPT(18)	4364177513707979689	0 28 42 54 64 88 130 142 144 208 210 222 264 288 298 310 324 352	-972645075970
STPT(10)	4364177550347147129	0 2 42 44 60 62 78 80 120 122	-936005908530
STPT(12)	4364177789938313537	0 2 84 86 114 116 150 152 180 182 264 266	-696414742122
SPT(16)	4364178148502150903	0 26 80 126 138 176 224 246 338 360 408 446 458 504 558 584	-337850904756
SPT(16)	4364178517101457363	0 24 30 36 46 58 60 64 150 154 156 168 178 184 190 214	30748401704
SPT(16)	4364178616741423621	0 16 36 42 60 82 126 130 138 142 186 208 226 232 252 268	130388367962
STPT(10)	4364178929531321777	0 2 30 32 60 62 90 92 120 122	443178266118
SPT(18)	4364179063305226247	0 36 84 90 104 132 134 140 162 164 186 192 194 222 236 242 290 326	576952170588
SPT(16)	4364179462237786717	0 16 54 66 70 96 114 120 130 136 154 180 184 196 234 250	975884731058