Triplet of three numbers

A weird combination of Cousin primes and Twin primes that are siblings, but known by their neighbor.... I shall dub these Alabama primes.

say "{.[0]+1}: ",$_ for grep *.all.is-prime, ^6000 .race.map: { $_-1, $_+3, $_+5 };

Output:

8: (7 11 13)
14: (13 17 19)
38: (37 41 43)
68: (67 71 73)
98: (97 101 103)
104: (103 107 109)
194: (193 197 199)
224: (223 227 229)
278: (277 281 283)
308: (307 311 313)
458: (457 461 463)
614: (613 617 619)
824: (823 827 829)
854: (853 857 859)
878: (877 881 883)
1088: (1087 1091 1093)
1298: (1297 1301 1303)
1424: (1423 1427 1429)
1448: (1447 1451 1453)
1484: (1483 1487 1489)
1664: (1663 1667 1669)
1694: (1693 1697 1699)
1784: (1783 1787 1789)
1868: (1867 1871 1873)
1874: (1873 1877 1879)
1994: (1993 1997 1999)
2084: (2083 2087 2089)
2138: (2137 2141 2143)
2378: (2377 2381 2383)
2684: (2683 2687 2689)
2708: (2707 2711 2713)
2798: (2797 2801 2803)
3164: (3163 3167 3169)
3254: (3253 3257 3259)
3458: (3457 3461 3463)
3464: (3463 3467 3469)
3848: (3847 3851 3853)
4154: (4153 4157 4159)
4514: (4513 4517 4519)
4784: (4783 4787 4789)
5228: (5227 5231 5233)
5414: (5413 5417 5419)
5438: (5437 5441 5443)
5648: (5647 5651 5653)
5654: (5653 5657 5659)
5738: (5737 5741 5743)