# Sum of digits of divisors

Psd(x) denotes the sum of digits of all positive integer divisors of x where x>0. For example, Psd(15)=1+3+5+1+5=14 (divisors: 1,3,5,15)

What is x for the biggest value of Psd(x) where x<10000.

• I would bet on 7560 or 9240, both of which have 64 divisors. I don't see a good way to be certain without a brute force search. – Ross Millikan Jan 17 '16 at 23:48
• So you're looking for the largest perfect number <10000? – Patrick Cook Jan 18 '16 at 6:52
• no, function is not exactly largest perfect number. – Oray Jan 18 '16 at 6:55
• Maybe a program could solve this – Daedric Jan 18 '16 at 10:33

Here's a simple Python bruteforce program that gets the job done:

from math import sqrt

maxNum = -1
maxSum = 0
for x in range(1, 10000):
currentSum = 0
for d in range(1, int(sqrt(x)) + 1):
if(x % d == 0):
for digit in str(d):
currentSum += int(digit)
for digit in str(x / d):
currentSum += int(digit)
if currentSum > maxSum:
maxSum = currentSum
maxNum = x

print "Maximum sum: {}.\n{}:".format(maxSum, maxNum)
for d in range(1, int(sqrt(maxNum)) + 1):
if(maxNum % d == 0):
print "  = {} x {}".format(d, maxNum / d)


Output:

Maximum sum: 549.
7920:
= 1 x 7920
= 2 x 3960
= 3 x 2640
= 4 x 1980
= 5 x 1584
= 6 x 1320
= 8 x 990
= 9 x 880
= 10 x 792
= 11 x 720
= 12 x 660
= 15 x 528
= 16 x 495
= 18 x 440
= 20 x 396
= 22 x 360
= 24 x 330
= 30 x 264
= 33 x 240
= 36 x 220
= 40 x 198
= 44 x 180
= 45 x 176
= 48 x 165
= 55 x 144
= 60 x 132
= 66 x 120
= 72 x 110
= 80 x 99
= 88 x 90


Good guess @Ross Millikan.

• @Oray I barely released my mouse button before you accepted it. Nice reflexes lol. – SpiritFryer Jan 18 '16 at 12:33