Integer n such sum of its aliquot divisors is a square

 

Some numbers never come up as aliquot sums,   such as 5.
They are called nonaliquot numbers or untouchable numbers
https://en.wikipedia.org/wiki/Untouchable_number

 
 
 

Let’s find the lowest   n   such that   s(n) = \sigma(n) - n = m^2,   2^2 \leq m^2 \leq 100^2

 
2^2 \; = \; 4 \; = \; \sigma(9) \; - \; 9 \; = \; 13 \; - \; 9
3^2 \; = \; 9 \; = \; \sigma(15) \; - \; 15 \; = \; 24 \; - \; 15

4^2 \; = \; 16 \; = \; \sigma(12) \; - \; 12 \; = \; 28 \; - \; 12
5^2 \; = \; 25 \; = \; \sigma(95) \; - \; 95 \; = \; 120 \; - \; 95
6^2 \; = \; 36 \; = \; \sigma(24) \; - \; 24 \; = \; 60 \; - \; 24
7^2 \; = \; 49 \; = \; \sigma(75) \; - \; 75 \; = \; 124 \; - \; 75
8^2 \; = \; 64 \; = \; \sigma(56) \; - \; 56 \; = \; 120 \; - \; 56
9^2 \; = \; 81 \; = \; \sigma(147) \; - \; 147 \; = \; 228 \; - \; 147

10^2 \; = \; 100 \; = \; \sigma(124) \; - \; 124 \; = \; 224 \; - \; 124
11^2 \; = \; 121 \; = \; \sigma(1199) \; - \; 1199 \; = \; 1320 \; - \; 1199
12^2 \; = \; 144 \; = \; \sigma(90) \; - \; 90 \; = \; 234 \; - \; 90
13^2 \; = \; 169 \; = \; \sigma(363) \; - \; 363 \; = \; 532 \; - \; 363
14^2 \; = \; 196 \; = \; \sigma(176) \; - \; 176 \; = \; 372 \; - \; 176
15^2 \; = \; 225 \; = \; \sigma(507) \; - \; 507 \; = \; 732 \; - \; 507
16^2 \; = \; 256 \; = \; \sigma(332) \; - \; 332 \; = \; 588 \; - \; 332
17^2 \; = \; 289 \; = \; \sigma(1075) \; - \; 1075 \; = \; 1364 \; - \; 1075

18^2   :    nonaliquot number

19^2 \; = \; 361 \; = \; \sigma(935) \; - \; 935 \; = \; 1296 \; - \; 935
20^2 \; = \; 400 \; = \; \sigma(524) \; - \; 524 \; = \; 924 \; - \; 524

 
other nonaliquot numbers :

24^2 = 576      28^2 = 784     36^2 = 1296     48^2 = 2304
50^2 = 2500    52^2 = 2704    56^2 = 3136    66^2 = 4356
72^2 = 5184    78^2 = 6084    84^2 = 7056    90^2 = 8100   
96^2 = 9216

 
 

Complete the list
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 

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About benvitalis

math grad - Interest: Number theory
This entry was posted in Number Puzzles and tagged . Bookmark the permalink.

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