# Problem 3

# mcdio takes an integer, num, and determines if it has a solution
# to the equation 20a + 9b + 6c = num. If it does, it returns True.
# If not, it returns nothing.
def mcdio(num):
    for i in range(0,num/20+1):
        for j in range(0,num/9+1):
	    for k in range(0,num/6+1):
	        if 20*i + 9*j + 6*k == num:
		    return True
		    print i,j,k
		    break

# mcdioSoln starts at 0 and tests every integer x to determine
# if there is a solution to 20a + 9b + 6c = x. When 5 consecutive
# x's are found such that there is a solution to the equation, it
# returns the last x such that there is no such solution.
def mcdioSoln():
    num = 0
    cons = 0
    bestFound = 0
    while cons < 6:
        num += 1
        if mcdio(num) == True:
	    cons += 1
	else:
	    cons = 0
	    bestFound = num
    return bestFound



# Problem 4

# gen_mcdio will be called by gen_mcdio_search to determine if the
# current number being tested by gen_mcdio_search has a solution
# for the given a,b,c.
def gen_mcdio(a,b,c,n):
    for i in range(0,n/c+1):
        for j in range(0,n/b+1):
	    for k in range(0,n/a+1):
	        if c*i + b*j + a*k == n:
	            return True

# gen_mcdio_search iterates through the positive integers less than
# 200 and calls gen_mcdio to test if a number has a solution.
def gen_mcdio_search(num1,num2,num3):
    cons = 0
    bestFound = 0
    while cons < 6:
        for n in range(0,200):
	    n += 1
            if gen_mcdio(num1,num2,num3,n):
	        cons += 1
	    else:
	        cons = 0
		bestFound = n
    return bestFound

# Example:
gen_mcdio_search(5,8,10)  # returns 27

# Problem 1

def is_prime(n):
    divisors = []
    for i in range(1, n/2+1):
        if n%i == 0:
	    divisors.append(i)
    if divisors == [1]:
	return True
    else:
	return False

def nth_prime(num):
    current = 3
    primesFound = 1
    assert num != 1
    while primesFound < num:
        if is_prime(current):
	    if primesFound == num-1:
	        break
	    current += 2
	    primesFound += 1
	else:
	    current += 2
    return current



# Problem 2

from math import *

def prime_log_sums(num):
    lSum = 0
    for i in range(1, num,2):
        if is_prime(i):
	    lSum = lSum + log(i)
    lSum = lSum + log(2)
    return lSum, num, lSum/num