Puzzle – Sum of aliquot divisors of n

 
 
The aliquot divisors of   n   are the divisors of   n   less than   n
 
 

Find two squares such that if each is increased by the sum of its aliquot divisors the resulting sums are equal.

for example,

16   has 5 divisors:     1,   2,   4,   8,   16
25   has 3 divisors:     1,   5,   25

(1 + 2 + 4 + 8)   +   16   =   (1 + 5)   +   25   =   31

 
Find other pairs.

 
 

And, here’s a set of three squares having the same sum of divisors

245828^2 \; = \; 60431405584
294867^2 \; = \; 86946547689
307285^2 \; = \; 94424071225

The sum is,   133151753133

 
Find other sets
 
 
 

 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
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One Response to Puzzle – Sum of aliquot divisors of n

  1. paul says:

    For part a, pairs

    4^2 & 5^2 Have sums of divisors = to 31
    12^2 & 15^2 Have sums of divisors = to 403
    28^2 & 35^2 Have sums of divisors = to 1767
    36^2 & 45^2 Have sums of divisors = to 3751
    44^2 & 55^2 Have sums of divisors = to 4123
    52^2 & 65^2 Have sums of divisors = to 5673
    68^2 & 85^2 Have sums of divisors = to 9517
    76^2 & 95^2 Have sums of divisors = to 11811
    84^2 & 105^2 Have sums of divisors = to 22971
    92^2 & 115^2 Have sums of divisors = to 17143
    108^2 & 135^2 Have sums of divisors = to 33883
    116^2 & 145^2 Have sums of divisors = to 27001
    124^2 & 155^2 Have sums of divisors = to 30783
    132^2 & 165^2 Have sums of divisors = to 53599
    148^2 & 185^2 Have sums of divisors = to 43617
    156^2 & 195^2 Have sums of divisors = to 73749

    For sets of 3

    1284^2 & 1528^2 & 1605^2 Have sums of divisors = to 4657471
    1628^2 & 1630^2 & 2035^2 Have sums of divisors = to 5801061
    1956^2 & 2030^2 & 2445^2 Have sums of divisors = to 10773399
    4884^2 & 4890^2 & 6105^2 Have sums of divisors = to 75413793
    5064^2 & 6172^2 & 7715^2 Have sums of divisors = to 73854183
    6216^2 & 7714^2 & 9291^2 Have sums of divisors = to 132408549
    8988^2 & 10696^2 & 11235^2 Have sums of divisors = to 265475847
    9768^2 & 12122^2 & 17441^2 Have sums of divisors = to 308953281
    10360^2 & 12388^2 & 15485^2 Have sums of divisors = to 315743463
    11396^2 & 11410^2 & 14245^2 Have sums of divisors = to 330660477
    11836^2 & 14397^2 & 14795^2 Have sums of divisors = to 299457613
    14124^2 & 16808^2 & 17655^2 Have sums of divisors = to 619443643
    14652^2 & 14670^2 & 18315^2 Have sums of divisors = to 701928381
    15544^2 & 15836^2 & 19795^2 Have sums of divisors = to 504081669

    Paul.

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