# For the Christmas tree, note that you can create a large number of
#   duplicates of a string by multiplying by a number.
#   For instance: 'ab'*4  yields: 'abababab'

def ChristmasTree(treeChar,edgeChar,height):
    print ' '*height+treeChar
    for i in range(1,height):
        print ' '*(height-i)+edgeChar+treeChar*(2*i-1)+edgeChar
    for i in range(2):
        print ' '*(height-1)+treeChar*3
        
def perfectNumbers(maxn):
    from math import sqrt
    for i in range(2,maxn+1):
        ifPerfectPrint(i)

# assume that n is a perfect number (it may not be) and prepare to
#   print it out if it is, but don't do it if it isn't
def ifPerfectPrint(n):
    from math import sqrt
    factsum=1
    s=str(n)+' = 1'
    sq=int(sqrt(n))
    for i in range(2,sq+1):
        if n%i==0:
            factsum+=i+n/i
            s+=' + '+str(i)+' + '+str(n/i)
    if factsum==n:
        print s