Explanation:

To begin with, all prime numbers will be part of this list.
There are 25 primes less than 100. (That is a nugget that can come in handy)
Apart from this, any number that can be written as a product of two or more primes will be there on this list.
That is, any number of the form pq, or pqr, or pqrs will be there on this list (where p, q, r, s are primes).
A number of the form pⁿq cannot be a part of this list if n is greater than 1, as then the number will be a multiple of p².
This is a brute-force question.
First let us think of all multiples of 2 * prime number. This includes 2 * 3, 2 * 5, 2 * 7, 2 * 11 all the way up to 2 * 47 (14 numbers).
The, we move on to all numbers of the type 3 * prime number 3 * 5, 3 * 7 all the way up to 3 * 31 (9 numbers).
Then, all numbers of the type 5 * prime number – 5 * 7, 5 * 11, 5 * 13, 5 * 17, 5 * 19 (5 numbers).
Then, all numbers of the type 7 * prime number and then 7 * 11, 7 * 13 (2 numbers).
There are no numbers of the form 11 * prime number which have not been counted earlier.
Post this, we need to count all numbers of the form p * q * r, where p, q, r are all prime.
In this list, we have 2 * 3 * 5, 2 * 3 * 7, 2 * 3 * 11, 2 * 3 * 13 and 2 * 5 * 7.
Adding 1 to this list, we get totally 36 different composite numbers.
Along with the 25 prime numbers, we get 61 numbers that cannot be written as a product of a perfect square greater than 1.