Problems:

1) Find the number d(n) of divisors of n for 1 <=n <= 200.

2) Find all the primes smaller than 1000.

3) Find all integers smaller than 1000 that are either perfect squares or sums of two perfect
   squares.

4) Find all integers smaller than 1000 that are either perfect squares or sums of two or three
   perfect squares.

5) Find all integers smaller than 1000 that are either perfect squares or sums of two, three or
   four perfect squares.

6) A number n is called a _quadratic residue modulo p_ (where p is a prime) if p DND n and if 
   there exists an integer m (1 <= m < p) such that p | m^2 - n. For p=11, find all numbers 
   r (1 <=r < p) thst are quadratic residues modulo p. Do the same for p=3, 5, 7, 17, 29 and 61.

7) Using the results of exercise 6, find the number of consecutive pairs of quadratic residues
   modulo p for p=3, 5, 7, 11, 17, 29 and 61. Compare your answer with p/4 in each case.

8) Find the number of ways an integer n can be represented as a sum of distinct positive
   integers for n <= 30 (for example, 5 can be represented in three ways: 5, 4+1 and 3+2).

9) Find the number of ways an integer n (n <= 30) can be represented as a sum of odd positive
   integers (for example, 5 can be represented in three ways: 5, 3+1+1 and 1+1+1+1+1). Compare
   your results with those obtained for exercise 8.
 
10) Find the sum sigma(n) of the divisors of n for 1 <= n <= 200

11) Let phi(n) denote the number of positive integers not exceeding n that are relatively prime
    to n; find phi(n) for 1 <= n <= 100.

12) Let b(n) denote the number of ways of representing n as a sum of nonnegative powers of 2 (for
    example, b(4) = 4 since 4 has the representations 4, 2+2, 2+1+1, 1+1+1=1); find b(n) for
    1 <= n <= 40. Are there any significant relationships between b(2n+1), b(2n), b(n), and
    b(n-1)?

13) For 1 <= n <= 15, compute the number t(n) of representations of n as a sum of positive 
    integers that are not multiples of 3 (for example, t(4) = 4 since 4 has the representations
    4, 2+2, 2+1+1, 1+1+1+1).

14) For 1 <= n <= 15, compute the number v(n) of representations of n as a sum of positive 
    integers in which no summand appears more than twice (for example, v(4) = 4, since 4 has the
    representations 4, 3+1, 2+2, 2+1+1). Compare v(n) with t(n).