There is a large literature on Hooley's $$ \Delta(n)=\max_u\sum_{d|n,\ e^u\le d< e^{u+1}}1 $$ giving its normal and average order. What is known of its maximal order?

Clearly $\Delta(n)\le d(n)$ and so the usual upper bounds apply, but these are not very close. By exhaustive calculation I have determined that $\Delta(n)\le138$ for $n\le5\cdot10^8$ with equality only for $465585120=2^5\cdot3^2\cdot5\cdot7\cdot11\cdot13\cdot 17\cdot19.$

# References

Paul Erdős, On abundant-like numbers, Canad. Math. Bull. 17 (1974), pp. 599-602.

C. Hooley, On a new technique and its applications to the theory of numbers, Proc. London Math. Soc. 3 38:1 (1979), pp. 115-151.

Helmut Maier and Gérald Tenenbaum, On the set of divisors of an integer, Invent. Math. 76 (1984), pp. 121-128.

Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors, J. London Math. Soc. 2 31:3 (1985), pp. 393-400.

Helmut Maier and Gérald Tenenbaum, On the normal concentration of divisors. II., Math. Proc. Cambridge Philos. Soc. 147:3 (2009), pp. 513-540.

Gérald Tenenbaum, Sur la concentration moyenne des diviseurs, Commentarii Mathematici Helvetici 60:1 (1985), pp. 411-428.