Problem $13.($Counting Primes$)$ Write a program called prime$\_$counter.py that accepts $n($int$)$ as command$-$line argument, and

writes to standard output the number of primes less than or equal to $n.$

$-/$vorkspace$/$controlf$1$ow$-$programs

python$3$ prime$\_$counter,py $10$

$4$

$8$ python$3$ prime$\_$counter.py $100$

$25$

$ python$3$ prime$\_$counter, py $1000$

$168$

Problem $14.($Perfect Numbers$)$ A perfect number is a positive integer whose proper divisors add up to the number. For

example, $6$ is a perfect number since its proper divisors $1,2,$ and $3$ add up to $6.$ Write a program called perfect$\_$numbers.py that

accepts $n($int$)$ as command$-$line argument, and writes to standard output the perfect numbers that are less than or equal to

n$.$

$2\_-/$workspace$/$controlflou$-$prograns

python$3perfec{t}_n$umbers.$py10$

python$3perfec{t}_n$umbers.$py1000$

Problem $15.($Ramanujan Numbers$)$ Srinivasa Ramanujan was an Indian mathematician who became famous for his intuition

for numbers. When the English mathematician G$.$ H$.$ Hardy came to visit him one day, Hardy remarked that the number of

his taxi was $1729,$ a rather dull number. Ramanujan replied, $"$No$,$ Hardy! It is a very interesting number. It is the smallest

number expressible as the sum of two cubes in two different ways." Verify this claim by writing a program ramanujan$\_$numbers.py

that accepts $n($int$)$ as command$-$line argument, and writes to standard output all integers less than or equal to $n$ that can

be expressed as the sum of two cubes in two different ways. In other words, find distinct positive integers $a,b,c,$ and $d$ such

that $a^3+b^3=c^3+d^{3}n$

$-/$Horzspace$/$controlt$10$y$-$programs

$8$ python$3$ ramanujan$\_$numbers. py $10000$

$1729=1-3+12-3=9-3+10-3$

$4104=2-3+163+15*3$

$ python$3$ ramanujan numbers. py $40000$

$13832=2-3+24-3=18-3+20-3$

$39312=2-3+34-3=15-3+33-3$

$32832=4-3+32-3=+18-3+30-3$

$20683=10\%3+27=3=19$:$3+24*3$