## 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