Uiua, 16 bytes

=⊸⁅⍜°√/+⊂⊚=0⊸◿⊸⇡

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⊸⇡     # range up to n
⊚=0⊸◿ # where mod by n equals 0
⊂     # join n
⍜°√/+ # under squaring, sum
=⊸⁅    # equal to rounded?

AWK, 42 bytes

{for(;i++<$1;)0~$1/i%1&&a+=i^2}$0=0~a^.5%1

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Prints one if truthy, nothing for falsey.

TI-BASIC (TI-84 Plus CE Python), 20 bytes

not(fPart(√(Σ(F²not(fPart(Ans/F)),F,1,Ans

J-uby, 77 bytes

~:^%(:& &(:%|:>&1)|:+&:select|:|&:+|~:|&(:sum+(~:**&2)|~:**&0.5|~:%&1|:==&0))

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CASIO BASIC (CASIO fx-9750GIII), 24 bytes

?→N
Not Frac √Σ(F²Not Frac N⌟F,F,1,N

outputs 1 or 0 for true or false respectively

Ruby, 45 41 bytes

-4 bytes from Jordan.

->n{(1..n).sum{n%_1<1?_1*_1:0}**0.5%1==0}

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Vyxal 3, 6 bytes

K²∑√/⌊

Vyxal It Online!

K²∑√/⌊­⁡​‎‎⁡⁠⁤‏‏​⁡⁠⁡‌⁢​‎‎⁡⁠⁣‏‏​⁡⁠⁡‌⁣​‎‎⁡⁠⁡‏‏​⁡⁠⁡‌⁤​‎‎⁡⁠⁢‏‏​⁡⁠⁡‌⁢⁡​‎‎⁡⁠⁢⁡‏⁠‎⁡⁠⁢⁢‏‏​⁡⁠⁡‌­
   √    # ‎⁡is the square root of
  ∑     # ‎⁢the sum of
K       # ‎⁣the implicit input's factors
 ²      # ‎⁤squared
    /⌊  # ‎⁢⁡an integer?
💎

Created with the help of Luminespire.

<script type="vyxal3">
K²∑√/⌊
</script>
<script>
    args=[["42"],["1"],["246"],["10"],["16"]]
</script>
<script src="https://themoonisacheese.github.io/snippeterpreter/snippet.js" type="module"/>

ARBLE, 33 bytes

nt(sqrt(1+sum(factors(n)|x^2))%1)

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Vyxal, 14 bytes (Test cases)

ɾḊT›ƛ:*;ƒ+√:ṙ=

Arturo, 39 bytes

$=>[=floor<=sqrt∑map factors&'x->x*x]

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$=>[              ; a function where input is assigned to &
    factors&      ; divisors of the input
    map _ 'x->x*x ; square each
    ∑             ; sum
    sqrt          ; square root
    <=            ; duplicate
    floor         ; floor
    =             ; are the equal?
]                 ; end function

Vyxal, 5 bytes

K²∑∆²

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   ∆² # Is
  ∑   # The sum of
 ²    # The squares of
K     # The factors of the input
   ∆² # A perfect square?

Thunno 2, 5 bytes

F²SƲ

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Explanation

F²SƲ  # Implicit input
F      # Factors
 ²     # Square
  S    # Sum
   Ʋ  # Perfect square
       # Implicit output

C# (Visual C# Interactive Compiler), 141 137 135 131 61 bytes

Thanks ASCII-only.

a=>Math.Sqrt(Enumerable.Range(1,a).Sum(b=>a%b==0?b*b:0))%1==0

Explanation:

a => Math.Sqrt(                                                    ) % 1 == 0 //Square root, test for whole number
               Enumerable.Range(1, a).Sum(                        ) //Sum 1 through a with..
                                          b => a % b==0 ? b * b: 0 //b^2 if a modulo b = 0, else 0

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Stax, 10 9 bytes

▀ùj╦┬44eR

Run and debug it at staxlang.xyz!

Unpacked (11 bytes) and explanation:

:d{J+kc|qJ=
:d             Implicit input. Make list of divisors.
  {  k         Reduce using block:
   J+            Square and add to the running total.
      c        Copy this sum of squares on the stack.
       |qJ     Take square root, rounding down. Square.
          =    Check for equality.

MATL, 9 bytes

Z\UsX^tk=

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As simple as it gets

Z\ % Divisors of (implicit) input
U  % Square
s  % Sum
X^ % Square root
t  % Duplicate this value
k= % Is it equal to its rounded value?

Pyth, 26 16 bytes

!%@s^R2*M{yPQ2 1

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I'm new to Pyth (and golfing in general) so this is pretty bad. Pyth doesn't have a built-in to get divisors, so I had to do that. Up until the perfect-square check, it was pretty alright, but after that it became a bit cluttered because of the if statement. Criticism is welcome.

-10 bytes thanks to Sok

Explanation:

            Q               #get user input
       *M{yP                #get the divisors
    ^R2                     #square each divisor
   s                        #sum the list
  @          2              #get square root
 %             1            #check if sqrt is a whole number (mod 1)
!                           #invert (0 -> True, any other num -> False)

Taxi, 4430 4305 bytes

Go to Post Office:w 1 l 1 r 1 l.Pickup a passenger going to The Babelfishery.Go to The Babelfishery:s 1 l 1 r.Pickup a passenger going to Cyclone.0 is waiting at Starchild Numerology.0 is waiting at Starchild Numerology.Go to Starchild Numerology:n 1 l 1 l 1 l 2 l.Pickup a passenger going to Rob's Rest.Pickup a passenger going to Addition Alley.Go to Rob's Rest:w 1 r 2 l 1 r.Go to Cyclone:s 1 l 1 l 2 l.[B]Pickup a passenger going to Divide and Conquer.1 is waiting at Starchild Numerology.Go to Starchild Numerology:n 1 r 3 l.Pickup a passenger going to Addition Alley.Go to Addition Alley:w 1 r 3 r 1 r 1 r.Pickup a passenger going to Cyclone.Go to Cyclone:n 1 l 1 l.Pickup a passenger going to Sunny Skies Park.Pickup a passenger going to Divide and Conquer.Go to Divide and Conquer:n 2 r 2 r 1 r.Pickup a passenger going to Cyclone.Go to Sunny Skies Park:e 1 l 1 l 2 l 1 l.Go to Zoom Zoom:n 1 r.Go to Cyclone:w.Pickup a passenger going to Sunny Skies Park.Pickup a passenger going to Equal's Corner.Pickup a passenger going to Trunkers.Go to Sunny Skies Park:n 1 r.Go to Trunkers:s 1 l.Pickup a passenger going to Equal's Corner.Go to Equal's Corner:w 1 l.Switch to plan C if no one is waiting.Pickup a passenger going to Riverview Bridge.Go to Sunny Skies Park:n.Pickup a passenger going to Sunny Skies Park.Pickup a passenger going to Cyclone.Go to Riverview Bridge:n 1 r 1 r.Go to Cyclone:w 2 l.Pickup a passenger going to Cyclone.Go to Sunny Skies Park:n 1 r.Go to Cyclone:n 1 l.Pickup a passenger going to Multiplication Station.Pickup a passenger going to Multiplication Station.Go to Multiplication Station:s 1 l 2 r 4 l.Pickup a passenger going to Addition Alley.Go to Rob's Rest:s 1 r 2 l 1 l 1 r 1 r.Pickup a passenger going to Addition Alley.Go to Addition Alley:s 1 l 1 l 2 r 1 r 1 r.Pickup a passenger going to Rob's Rest.Go to Rob's Rest:n 1 l 1 l 1 l 2 r 1 r.Go to Cyclone:s 1 l 1 l 2 l.Pickup a passenger going to Magic Eight.Go to Sunny Skies Park:n 1 r.Pickup a passenger going to Cyclone.Go to Cyclone:n 1 l.Pickup a passenger going to Cyclone.Pickup a passenger going to Magic Eight.Go to Magic Eight:s 1 l 2 r.Switch to plan D if no one is waiting.Pickup a passenger going to Addition Alley.Go to Cyclone:n 1 l 2 r.Switch to plan B.[C]Go to Sunny Skies Park:n.Pickup a passenger going to Cyclone.Pickup a passenger going to Addition Alley.Go to Cyclone:n 1 l.Switch to plan B.[D]Go to Cyclone:n 1 l 2 r.Pickup a passenger going to Sunny Skies Park.Pickup a passenger going to Sunny Skies Park.Go to Sunny Skies Park:n 1 r.Go to Rob's Rest:s 2 r 1 r.Pickup a passenger going to Cyclone.Go to Cyclone:s 1 l 1 l 2 l.Pickup a passenger going to Cyclone.Pickup a passenger going to Cyclone.Go to Zoom Zoom:n.Go to Cyclone:w.[E]Pickup a passenger going to Joyless Park.Pickup a passenger going to Divide and Conquer.Go to Joyless Park:n 2 r 2 r 2 l.Go to Cyclone:w 1 r 2 l 2 l.Pickup a passenger going to Cyclone.Pickup a passenger going to Joyless Park.Go to Joyless Park:n 2 r 2 r 2 l.Go to Cyclone:w 1 r 2 l 2 l.Pickup a passenger going to Divide and Conquer.Pickup a passenger going to Divide and Conquer.Go to Divide and Conquer:n 2 r 2 r 1 r.Pickup a passenger going to Magic Eight.1 is waiting at Starchild Numerology.Go to Starchild Numerology:e 1 r 3 r 1 l 1 l 2 l.Pickup a passenger going to Magic Eight.Go to Magic Eight:w 1 r 2 r 1 r.Switch to plan F if no one is waiting.Pickup a passenger going to Riverview Bridge.Go to Joyless Park:e 2 l 4 r.Pickup a passenger going to Cyclone.Pickup a passenger going to The Underground.Go to The Underground:w 1 l.Pickup a passenger going to Cyclone.Go to Fueler Up:s.Go to Riverview Bridge:n 3 l.Go to Cyclone:w 2 l.Switch to plan E.[F]Go to Joyless Park:e 2 l 4 r.Pickup a passenger going to What's The Difference.Pickup a passenger going to Cyclone.Go to Cyclone:w 1 r 2 l 2 l.Pickup a passenger going to Multiplication Station.Pickup a passenger going to Multiplication Station.Go to Multiplication Station:s 1 l 2 r 4 l.Pickup a passenger going to What's The Difference.Go to What's The Difference:n 2 l 1 r 3 l.Pickup a passenger going to Knots Landing.Go to Knots Landing:e 4 r 1 l.Pickup a passenger going to The Babelfishery.Go to The Babelfishery:w 1 l.Pickup a passenger going to Post Office.Go to Post Office:n 1 l 1 r.

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I went to a lot of places in Townsburg that I've never visited before for this program. Because this program is HUGE (mainly due to the fact that there isn't a one-stop way to take square roots; I still don't actually do that, though), I'm going to attempt to sketch out an explanation of this program.

Go to Post Office:w 1 l 1 r 1 l.

[Take an input line from STDIN.]

Pickup a passenger going to The Babelfishery.
Go to The Babelfishery:s 1 l 1 r.

[Take it to The Babelfishery, which converts the string to a double.]
[If you were to give it a double, it gives back a string.]

Pickup a passenger going to Cyclone.

[We're going to clone this number by sending it to the Cyclone.]

0 is waiting at Starchild Numerology.

[In Taxi, numeric passengers are introduced once they wait at Starchild Numerology.]
[This passenger will be our running squares total.]

0 is waiting at Starchild Numerology.

[This passenger will be our counter, going from 1 to the inputted number.]
[Yes, it's starting at 0 here, but it is incremented at the start of our loop.]

Go to Starchild Numerology:n 1 l 1 l 1 l 2 l.
Pickup a passenger going to Rob's Rest.
Pickup a passenger going to Addition Alley.
Go to Rob's Rest:w 1 r 2 l 1 r.
Go to Cyclone:s 1 l 1 l 2 l.

[Drop off the squares total at Rob's Rest. We'll pick it back up later when we need it.]
[Also, do the actual cloning of the inputted number.]

[At the start of each loop, our only passenger is the counter, going to Addition Alley.]

[B]
Pickup a passenger going to Divide and Conquer.

[Pick up one of the clones of the inputted number.]
[We will leave the other clone here for later.]

1 is waiting at Starchild Numerology.
Go to Starchild Numerology:n 1 r 3 l.
Pickup a passenger going to Addition Alley.
Go to Addition Alley:w 1 r 3 r 1 r 1 r.
Pickup a passenger going to Cyclone.

[Pick up the number 1, and add it to the counter, then pick up the result.]
[This new passenger is our new counter.]

Go to Cyclone:n 1 l 1 l.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Divide and Conquer.

[First we pick up the other clone of the inputted number, going to Sunny Skies Park.]
[Sunny Skies Park is basically a First-In-First-Out queue, where we will stick it for later.]
[Then we pick up a clone of the counter which we just dropped off, so we can divide the input by it.]
[We will leave the other clone here for later.]

Go to Divide and Conquer:n 2 r 2 r 1 r.
Pickup a passenger going to Cyclone.

[Divide the inputted number by the counter, and prepare to clone it.]

Go to Sunny Skies Park:e 1 l 1 l 2 l 1 l.

[Drop off our clone of the inputted number at Sunny Skies Park.]

Go to Zoom Zoom:n 1 r.

[Can't forget to get gas!]
[Zoom Zoom is a close gas station, and we have so many credits that one stop here every iteration fills us up completely.]

Go to Cyclone:w.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Equal's Corner.
Pickup a passenger going to Trunkers.

[First we pick up the other clone of the counter, going to Sunny Skies Park for later.]
[Then we pick up a clone of the division result, going to Equal's Corner.]
[Equal's Corner takes multiple numeric passengers, and returns the value of one of them if they are equal, but otherwise it returns no one.]
[Finally we pick up the other clone of the division result, going to Trunkers.]
[Dropping off a passenger at Trunkers is equivalent to the floor function. We need it here because division results are "exact".]
[(Well, as exact as double precision arithmetic can be.)]
[We will compare the result of this to the exact result of the division.]

Go to Sunny Skies Park:n 1 r.

[Drop off our clone of the counter at Sunny Skies Park.]

Go to Trunkers:s 1 l.
Pickup a passenger going to Equal's Corner.
Go to Equal's Corner:w 1 l.

[Compare the exact division result with the floored division result.]
[If they are equal, then the value of one of them will be waiting.]
[This also means that the counter is a divisor of the input, and we should add the square of the counter to our running squares total.]
[If they are not equal, no one will be waiting, the counter is not a divisor of the input, and nothing should happen.]

Switch to plan C if no one is waiting.

[This is Taxi's only conditional operator.]
[It jumps to a bracketed label (in this case, C) if there is no one waiting at this stop.]
[Of course, there is an unconditional version of this, too.]

[If we are on this path, we are adding the square of the counter to our running squares total.]

Pickup a passenger going to Riverview Bridge.

[We don't actually need the passenger that's waiting here (and leaving it there will not work at all).]
[So what do you do when you have a pesky passenger you don't want? You take them to Riverview Bridge!]
[Riverview Bridge has a lovely view, but passengers dropped off there always seem to fall in the river.]
[This means you don't collect your fare for bringing them there, but at least the pesky passenger is gone.]

Go to Sunny Skies Park:n.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Cyclone.

[We want to pick up the counter (so we can multiply it by itself). That dang input is in the way, but we still need it.]
[So what do we do? We just take it back to where it was waiting!]
[Yup, even though one would never do this in real life, a passenger's current location is a perfectly legal destination in Taxi.]

Go to Riverview Bridge:n 1 r 1 r.

[So long, Equal's Corner result!]

Go to Cyclone:w 2 l.
Pickup a passenger going to Cyclone.
Go to Sunny Skies Park:n 1 r.

[We clone the counter so we can still use it, then we drop the inputted number back at Sunny Side Park.]
[Again, we will leave the other clone here for later.]

Go to Cyclone:n 1 l.
Pickup a passenger going to Multiplication Station.
Pickup a passenger going to Multiplication Station.
Go to Multiplication Station:s 1 l 2 r 4 l.

[Square the counter.]

Pickup a passenger going to Addition Alley.
Go to Rob's Rest:s 1 r 2 l 1 l 1 r 1 r.
Pickup a passenger going to Addition Alley.
Go to Addition Alley:s 1 l 1 l 2 r 1 r 1 r.
Pickup a passenger going to Rob's Rest.
Go to Rob's Rest:n 1 l 1 l 1 l 2 r 1 r.

[Pick up the running squares total from Rob's Rest, add it to the square of the counter, then bring it back.]

Go to Cyclone:s 1 l 1 l 2 l.
Pickup a passenger going to Magic Eight.

[Pick up the counter, going to Magic Eight.]
[Magic Eight takes two numeric passengers, and returns the first passenger if it is less than the second, but otherwise it returns no one.]
[We will compare the counter to the inputted number...]

Go to Sunny Skies Park:n 1 r.
Pickup a passenger going to Cyclone.
Go to Cyclone:n 1 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Magic Eight.

[...who we take now, so we can make a clone of it and compare it to the counter.]

Go to Magic Eight:s 1 l 2 r.
Switch to plan D if no one is waiting.

[Compare.]
[If the counter is waiting here, the loop is to continue.]
[If not, switch to plan D, end the loop, and start detecting whether or not the running squares total is a square itself.]

Pickup a passenger going to Addition Alley.
Go to Cyclone:n 1 l 2 r.
Switch to plan B.

[Reset everything, and loop back to the start of plan B.]

[C]

[If we are on this path, the counter is not a divisor of the inputted number, but the loop should continue anyways.]

Go to Sunny Skies Park:n.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Addition Alley.
Go to Cyclone:n 1 l.
Switch to plan B.

[Reset everything, and loop back to the start of plan B.]

[D]

[We are finally out of the loop. Now it's time for the real fun.]

Go to Cyclone:n 1 l 2 r.
Pickup a passenger going to Sunny Skies Park.
Pickup a passenger going to Sunny Skies Park.
Go to Sunny Skies Park:n 1 r.

[We don't need the inputted number anymore, but we can't change its destination from the Cyclone.]
[So we go to the Cyclone, so we can take the two clones and bring them to Sunny Skies Park forever.]
[We could send them to Riverview Bridge, but that will take more bytes.]

Go to Rob's Rest:s 2 r 1 r.

[Pick up our running squares total from Rob's Rest, so it can go out and see the town.]

Pickup a passenger going to Cyclone.
Go to Cyclone:s 1 l 1 l 2 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Cyclone.
Go to Zoom Zoom:n.
Go to Cyclone:w.

[We need to end up with 4 clones of the total.]
[Two of them will actually be used as the counter in this loop.]

[At the start of each loop, we have no passengers.]

[E]
Pickup a passenger going to Joyless Park.
Pickup a passenger going to Divide and Conquer.
Go to Joyless Park:n 2 r 2 r 2 l.

[We will leave one clone of the total at Joyless Park (like Sunny Side Park, but on the opposite side of town).]
[The other clone will be used in a division.]

Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Cyclone.
Pickup a passenger going to Joyless Park.
Go to Joyless Park:n 2 r 2 r 2 l.

[We will leave one clone of the counter at Joyless Park.]
[We will clone the other clone (clone-ception?).]

Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Divide and Conquer.
Pickup a passenger going to Divide and Conquer.
Go to Divide and Conquer:n 2 r 2 r 1 r.

[We will use both clones of the counter in a division.]
[Divide and Conquer can take 3 arguments, and the result here will be equivalent to total / counter^2 .]
[If the square of the counter is equal to the total, the result will be 1.]
[If the square of the counter is less than the total, the result will be greater than 1.]
[If the square of the counter is greater than the total, the result will be less than 1.]

Pickup a passenger going to Magic Eight.
1 is waiting at Starchild Numerology.
Go to Starchild Numerology:e 1 r 3 r 1 l 1 l 2 l.
Pickup a passenger going to Magic Eight.
Go to Magic Eight:w 1 r 2 r 1 r.
Switch to plan F if no one is waiting.

[We compare the result of the division against 1 to decide our next course of action.]
[If it is greater or equal, the loop should end, and the counter is equal to the floor of the square root.]
[If it is less, the loop should continue.]

Pickup a passenger going to Riverview Bridge.

[Whoops, we don't actually need this anymore.]

Go to Joyless Park:e 2 l 4 r.
Pickup a passenger going to Cyclone.

[We will clone the total later, so we can reuse it in the next loop iteration.]

Pickup a passenger going to The Underground.
Go to The Underground:w 1 l.

[Instead of taking the number 1 and subtracting it from the counter, we can do an optimization here.]
[The Underground is a destination that takes 1 passenger, and subtracts 1 from it.]
[If the result is positive, it returns the result. Otherwise, it returns no one.]
[The latter result will not happen for us, however.]

Pickup a passenger going to Cyclone.

[We will clone the counter later, so we can reuse it in the next loop iteration.]

Go to Fueler Up:s.

[Gotta remember to stop for gas!]

Go to Riverview Bridge:n 3 l.

[So long, Magic Eight result!]

Go to Cyclone:w 2 l.
Switch to plan E.

[Reset everything, and loop back to the start of plan E.]

[F]

Go to Joyless Park:e 2 l 4 r.
Pickup a passenger going to What's The Difference.
Pickup a passenger going to Cyclone.

[Once we've figured out the square root of the total, we're still not done.]

Go to Cyclone:w 1 r 2 l 2 l.
Pickup a passenger going to Multiplication Station.
Pickup a passenger going to Multiplication Station.
Go to Multiplication Station:s 1 l 2 r 4 l.

[We actually square the square root...]

Pickup a passenger going to What's The Difference.
Go to What's The Difference:n 2 l 1 r 3 l.

[...and subtract it from the total at What's The Difference (which subtracts dropped-off passengers).]

Pickup a passenger going to Knots Landing.
Go to Knots Landing:e 4 r 1 l.

[If the result is 0, the total is a perfect square, and a truthy value should be output.]
[If the result is nonzero, the total is not a perfect square, and a falsy value should be output.]
[Luckily, Knots Landing does our work for us.]
[Knots Landing inverts the boolean logic of numeric passengers (i.e. 0 becomes 1, nonzero becomes 0).]

Pickup a passenger going to The Babelfishery.
Go to The Babelfishery:w 1 l.

[Convert it to a string...]

Pickup a passenger going to Post Office.
Go to Post Office:n 1 l 1 r.

[...and output it.]

[We would exit the program by going to the Taxi Garage, but that takes bytes.]
[The boss fires us because we didn't bring the taxi back to the garage, but that gets output to STDERR, so that's OK.]

APL(NARS), 23 chars, 46 bytes

{0=1∣√+/×⍨(0=a∣⍵)/a←⍳⍵}

test:

   g←{0=1∣√+/×⍨(0=a∣⍵)/a←⍳⍵}     
   g¨ 42 1 246 10 16
1 1 1 0 0 

MathGolf, 5 4 bytes

─²Σ°

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Explanation

─     Get all divisors as list (implicit input)
 ²    Square (implicit map)
  Σ   Sum
   °  Is perfect square?

Very similar to other answers, compared to 05AB1E I gain one byte for my "is perfect square" operator.

C# (.NET Core), 115 74 bytes

Thanks to user Misha Lavrov for teaching me how to properly do code golf! :)

a=>{int s=0;for(int i=0;i++<=a;)s+=a%i==0?i*i:0;return Math.Sqrt(s)%1==0;}

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Ungolfed:

a => {                              // reads input
    int s = 0;                      // holds sum of squares
    for(int i = 0; i++ <= a;)       // index from 1 to input, inclusive
    {
        s +=                        // add to sum
            a % i == 0 ?            // check if remainder of input divided by current index is zero
                i * i : 0;          // if true, add square of current index, else add zero
    }
    return Math.Sqrt(s) % 1 == 0;   // prints "True" if square root is whole number, "False" otherwise
}

Ohm v2, 5 bytes

V²ΣƲ

Try it online! Explanation:

V²ΣƲ    
V      Pushes input's divisors
 ²     Squares
   Σ   Sum
    Ʋ Returns whether value is a proper square.

Wolfram Language (Mathematica), 24 20 bytes

1∣Norm@Divisors@#&

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Divisors finds all the divisors of a number, Norm takes the square root of the sum of squares, and then we test if this is divisible by 1 (i.e., is an integer).

Java 8, 75 70 bytes

n->{int s=0,i=0;for(;++i<=n;)s+=n%i<1?i*i:0;return Math.sqrt(s)%1==0;}

-5 bytes thanks to @archangel.mjj.

Try it online.

Explanation:

n->{             // Method with integer parameter and boolean return-type
  int s=0,       //  Sum-integer, starting at 0
      i=0;       //  Divisor integer, starting at 0
  for(;++i<=n;)  //  Loop `i` in the range [1, n]
    s+=n%i<1?    //   If `n` is divisible by `i`:
        i*i      //    Increase the sum by the square of `i`
       :         //   Else:
        0;       //    Leave the sum the same by adding 0
  return Math.sqrt(s)%1==0;}
                 //  Return whether the sum `s` is a perfect square

Groovy, 47 bytes

A lambda accepting a numeric argument.

n->s=(1..n).sum{i->n%i?0:i*i}
!(s%Math.sqrt(s))

Explanation

(1..n) creates an array of the values 1 to n

n%i is false (as 0 is falsy) if i divides n without remainder

n%i ? 0 : i*i is the sum of the square of the value i if it divides n without remainder, otherwise is 0

sum{ i-> n%i ? 0 : i*i } sums the previous result across all i in the array.

s%Math.sqrt(s) is false (as 0 is falsy) if the sqrt of s divides s without remainder

!(s%Math.sqrt(s)) returns from the lambda (return implicit on last statement) !false when the sqrt of s divides s without remainder

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Perl 5, 47 bytes

$a+=$_*$_*!($n%$_)for 1..$n;$a=!($a**.5=~/\D/); 

Returns 1 for true and nothing for false.

Explanation:

$a+=              for 1..$n;                      sum over i=1 to n
    $_*$_                                         square each component of the sum
         *!($n%$_)                                multiply by 1 if i divides n.
                            $a=                   a equals
                                ($a**.5           whether the square root of a
                               !       =~/\D/);   does not contain a non-digit.

Pyt, 7 bytes

ð²ƩĐř²∈

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Explanation

            Implicit input
ð           Get list of divisors
 ²          Square each element
  Ʃ         Sum the list [n]
   Đ        Duplicate the top of the stack
    ř²      Push the first n square numbers
      ∈     Is n in the list of square numbers?
            Implicit output

ð²Ʃ√ĐƖ=

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Explanation

            Implicit input
ð           Get list of divisors
 ²          Square each element
  Ʃ         Sum the list [n]
   √        Take the square root of n
    Đ       Duplicate the top of the stack
     Ɩ      Cast to an integer
      =     Are the top two elements on the stack equal to each other?
            Implicit output

ð²Ʃ√1%¬

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Explanation

            Implicit input
ð           Get list of divisors
 ²          Square each element
  Ʃ         Sum the list [n]
   √        Take the square root of n
    1%      Take the square root of n modulo 1
      ¬     Negate [python typecasting ftw :)]
            Implicit output

Brachylog, 12 8 bytes

f^₂ᵐ+~^₂

-4 bytes thanks to Fatelize cause i didn't realize brachylog has a factors functions

explanation

f^₂ᵐ+~^₂            #   full code
f                   #       get divisors
 ^₂ᵐ                #           square each one
    +               #       added together
      ~^₂           #       is the result of squaring a number

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F#, 111 bytes

let d n=Seq.where(fun v->n%v=0){1..n}
let u n=
 let m=d n|>Seq.sumBy(fun x->x*x)
 d m|>Seq.exists(fun x->x*x=m)

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So d gets the divisors for all numbers between 1 and n inclusive. In the main function u, the first line assigns the sum of all squared divisors to m. The second line gets the divisors for m and determines if any of them squared equals m.

JavaScript (ES7),  46 44  42 bytes

Saved 1 byte thanks to @Hedi

n=>!((g=d=>d&&d*d*!(n%d)+g(d-1))(n)**.5%1)

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Commented

n =>             // n = input
  !(             // we will eventually convert the result to a Boolean
    (g = d =>    // g is a recursive function taking the current divisor d
      d &&       //   if d is equal to 0, stop recursion 
      d * d      //   otherwise, compute d²
      * !(n % d) //   add it to the result if d is a divisor of n
      + g(d - 1) //   add the result of a recursive call with the next divisor
    )(n)         // initial call to g with d = n
    ** .5 % 1    // test whether the output of g is a perfect square
  )              // return true if it is or false otherwise

Perl 6, 34 bytes

-1 byte thanks to nwellnhof

{grep($_%%*,1..$_)>>².sum**.5%%1}

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Haskell, 78 64 53 bytes

-14 bytes thanks to Ørjan Johansen. -11 bytes thanks to ovs.

f x=sum[i^2|i<-[1..x],x`mod`i<1]`elem`map(^2)[1..x^2]

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Hey, it's been a while since I've... written any code, so my Haskell and golfing might a bit rusty. I forgot the troublesome Haskell numeric types. :P

Scala, 68 67 bytes

def j(s:Int)=Math.sqrt((1 to s).filter(s%_<1).map(a=>a*a).sum)%1==0

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C (gcc), 67 63 60 59 bytes

-1 bytes thanks to @JonathanFrech

i,s;f(n){for(s=i=0;i++<n;)s+=n%i?0:i*i;n=sqrt(s);n=n*n==s;}

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Red, 67 bytes

func[n][s: 0 repeat d n[if n % d = 0[s: d * d + s]](sqrt s)% 1 = 0]

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Matlab, 39 37 bytes

@(v)~mod(sqrt(sum(divisors(v).^2)),1)

Unfortunately, it doesn't work on Octave (on tio) so no tio link.

Note As @LuisMendo stated, divisors() belongs to Symbolic Toolbox.

Pari/GP, 23 bytes

n->issquare(sigma(n,2))

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K (oK), 26 25 22 bytes

Solution:

{~1!%+/x*x*~1!x%:1+!x}

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Explanation:

{~1!%+/x*x*~1!x%:1+!x} / the solution
{                    } / lambda taking x as input
                   !x  / range 0..x-1                        \
                 1+    / add 1                               |
              x%:      / x divided by and save result into x |
            1!         / modulo 1                            | get divisors
           ~           / not                                 |
         x*            / multiply by x                       /
       x*              / multiply by x (aka square)          > square
     +/                / sum up                              > sum up
    %                  / square root                         \  
  1!                   / modulo 1                            | check if a square
 ~                     / not                                 / 

Notes:

• -1 bytes taking inspiration from the PowerShell solution
• -3 bytes taking inspiration from the APL solution

Shakespeare Programming Language, 434 428 415 bytes

,.Ajax,.Ford,.Puck,.Act I:.Scene I:.[Enter Ajax and Ford]Ford:Listen tothy.Scene V:.Ajax:You be the sum ofyou a cat.Ford:Is the remainder of the quotient betweenyou I worse a cat?[Exit Ajax][Enter Puck]Ford:If soyou be the sum ofyou the square ofI.[Exit Puck][Enter Ajax]Ford:Be you nicer I?If solet usScene V.[Exit Ford][Enter Puck]Puck:Is the square ofthe square root ofI worse I?You zero.If notyou cat.Open heart

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-13 bytes thanks to Jo King!

Outputs 1 for true result, outputs 0 for false result.

Octave / MATLAB, 43 bytes

@(n)~mod(sqrt(sum(find(~mod(n,1:n)).^2)),1)

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Mathematica, 32 bytes

IntegerQ@Sqrt[2~DivisorSigma~#]&

Pure function. Takes a number as input and returns True or False as output. Not entirely sure if there's a shorter method for checking perfect squares.

APL (Dyalog Unicode), 18 bytes

0=1|.5*⍨2+.*⍨∘∪⍳∨⊢

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Anonymous lambda. Returns 1 for truthy and 0 for falsy (test cases in TIO are prettified).

Shoutouts to @H.PWiz for 4 bytes!

How:

0=1|.5*⍨2+.*⍨∘∪⍳∨⊢   ⍝ Main function, argument ⍵ → 42
                ∨⊢   ⍝ Greatest common divisor (∨) between ⍵ (⊢)
               ⍳      ⍝ and the range (⍳) [1..⍵]
              ∪      ⍝ Get the unique items (all the divisors of 42; 1 2 3 6 7 14 21 42)
             ∘        ⍝ Then
            ⍨         ⍝ Swap arguments of
        2+.*          ⍝ dot product (.) of sum (+) and power (*) between the list and 2 
                      ⍝ (sums the result of each element in the vector squared)
       ⍨              ⍝ Use the result vector as base
    .5*               ⍝ Take the square root
  1|                  ⍝ Modulo 1
0=                    ⍝ Equals 0

Proton, 41 bytes

a=>sum(q*q for q:1..a+1if a%q<1)**.5%1==0

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Similar approach to the Python answer.

Gaia, 5 bytes

ds¦Σụ

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Japt, 11 9 7 bytes

-2 bytes from @Giuseppe and another -2 from @Shaggy

â x²¬v1

â x²¬v1             Full program. Implicity input U
â                   get all integer divisors of U
  x²                square each element and sum
    ¬               square root result
     v1           return true if divisible by 1

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Jelly, 6 bytes

ÆD²SƲ

Try it online! Or see the test-suite.

How?

ÆD²SƲ - Main Link: integer
ÆD     - divisors
  ²    - square
   S   - sum
    Ʋ - is square?

05AB1E, 5 bytes

ÑnOŲ

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How?

ÑnOŲ
Ñ     - divisors
 n    - square
  O   - sum
   Ų - is square?

PowerShell, 68 56 bytes

param($n)1..$n|%{$a+=$_*$_*!($n%$_)};1..$a|?{$_*$_-eq$a}

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Seems long ...
-12 bytes thanks to mazzy

Does exactly what it says on the tin. Takes the range from 1 to input $n and multiplies out the square $_*$_ times whether it's a divisor or not !($n%$_). This makes divisors equal to a nonzero number and non-divisors equal to zero. We then take the sum of them with our accumulator $a. Next, we loop again from 1 up to $a and pull out those numbers where |?{...} it squared is -equal to $a. That is left on the pipeline and output is implicit.

Outputs a positive integer for truthy, and nothing for falsey.

Husk, 6 bytes

£İ□ṁ□Ḋ

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Explanation

£İ□ṁ□Ḋ  -- example input 12
     Ḋ  -- divisors: [1,2,3,4,6,12]
   ṁ    -- map the following ..
    □   -- | square: [1,4,9,16,36,144]
        -- .. and sum: 210
£       -- is it element of (assumes sorted)
 İ□     -- | list of squares: [1,4,9,16..196,225,..

Python 2, 55 bytes

lambda n:sum(i*i*(n%i<1)for i in range(1,n+1))**.5%1==0

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Neim, 5 bytes

𝐅ᛦ𝐬q𝕚

Explanation:

𝐅      Factors
 ᛦ      Squared
  𝐬     Summed
    𝕚   is in?
   q    infinite list of square numbers

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R, 39 37 bytes

!sum((y=1:(x=scan()))[!x%%y]^2)^.5%%1

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Uses the classic "test if perfect square" approach, taking the non-integral part of the square root S^.5%%1 and taking the logical negation of it, as it maps zero (perfect square) to TRUE and nonzero to FALSE.

Thanks to Robert S for saving a couple of bytes!