# Distribution and upper bound of mimic numbers

**Problem**

Let the notation denote '' divides ''. The mimic function in number theory is defined as follows [1].

**Definition**For any positive integer divisible by , the mimic function, , is given by,

By using this definition of mimic function, the mimic number of any non-prime integer is defined as follows [1].

**Definition**The number is defined to be the mimic number of any positive integer , with respect to , for the minimum value of which .

Given these two definitions and a positive integer , find the distribution of mimic numbers of those numbers divisible by .

Again, find whether there is an upper bound of mimic numbers for a set of numbers divisible by any fixed positive integer .

## Bibliography

*[1] Malay Bhattacharyya, Sanghamitra Bandyopadhyay and U Maulik, Non-primes are recursively divisible, Acta Universitatis Apulensis 19 (2009).

* indicates original appearance(s) of problem.