A298565 a(n) is the largest prime p congruent to 1 mod n such that the multiplicative subgroup H of (Z/pZ)* of index n contains no nontrivial mod-p arithmetic progression of length 3.

7, 19, 37, 31, 127, 197, 97, 181, 191, 463, 421, 937, 337, 1321, 881, 2347, 3889, 2699, 1861, 1009, 2861, 1979, 3793, 1951, 2861, 3889, 1933, 1973, 4831, 1861, 7393, 8053, 8297, 5881, 8209, 3257, 7753, 17551, 7841, 6971, 16843, 16943, 15401, 12781, 17573, 41549, 12097