Hooley's delta function
In mathematics, Hooley's delta function (), also called Erdős--Hooley delta-function, defines the maximum number of divisors of  in  for all , where  is the Euler's number. The first few terms of this sequence are
 . History
The sequence was first introduced by Paul Erdős in 1974, then studied by Christopher Hooley in 1979. In 2023, Dimitris Koukoulopoulos and Terence Tao proved that the sum of the first  terms, , for . In particular, the average order of  to  is  for any . Later in 2023 Kevin Ford, Koukoulopoulos, and Tao proved the lower bound , where , fixed , and . Usage 
This function measures the tendency of divisors of a number to cluster. The growth of this sequence is limited by  where  is the number of divisors of . See also 
 Divisor function
 Euler's number

References

Divisor function
Arithmetic functions
Number theory
Integer sequences