Jelly, 40 32 bytes

1SÆD>Ðḟ½ƊṀṭƲ³¡Äż$Ḷ:Ḃ^þ`ʋ/€Ṛṛ""/Y

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1. Generate a list of values of \$k\$ and the cumulative sum of \$k\$
2. Generate a checkerboard for each of these with the tile size of \$k\$ and the board size of the cumulative sum
3. Work backwards through the list of checkerboards, each time replacing the top-left section of the board with the new, smaller board.

Ṛ      | Reverse list
    /  | Reduce using the following:
 ṛ""   | - Replace the top left portion of the current matrix with the next one
     Y | Finally join by newlines

Jelly, 40 31 bytes

1SÆD>Ðḟ½ƊṀṭƲ³¡Äż$Ḷ:Ḃ^þ`ʋ/€ḷ""/Y

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1. Generate a list of values of \$k\$ and the cumulative sum of \$k\$
2. Generate a checkerboard for each of these with the tile size of \$k\$ and the board size of the cumulative sum
3. Work through the list of checkerboards, each time replacing the top-left section of the next board with the existing board.

   /  | Reduce using the following:
ḷ""   | - Replace the top left portion of the next matrix with the current one
    Y | Finally join by newlines

Jelly, 40 33 bytes

1SÆD>Ðḟ½ƊṀ;@Ʋ³¡Äż$Ḷ:Ḃ^þ`ʋ/€Ṛṛ""/Y

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1                  | Start with 1
            Ʋ³¡    | Loop through the following the number of times indicated by the first argument to the program; this generates a list of values of k
 S                 | - Sum
        Ɗ          | - Following three links as a monad 
  ÆD               |   - List of divisors
    >Ðḟ½           |   - Exclude those greater than the square root
         Ṁ         |   - Maximum
          ;@       | - Concatenate this to the end of the current list of values of k 
               Äż$ | Zip the cumulative sum of the values of k with the values

Jelly, 40 32 bytes

1SÆD>Ðḟ½ƊṀṭƲ³¡Äż$Ḷ:Ḃ^þ`ʋ/€Ṛṛ""/Y

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1                 | Start with 1
           Ʋ³¡    | Loop through the following the number of times indicated by the first argument to the program; this generates a list of values of k
 S                | - Sum
        Ɗ         | - Following three links as a monad 
  ÆD              |   - List of divisors
    >Ðḟ½          |   - Exclude those greater than the square root
         Ṁ        |   - Maximum
          ṭ       | - Concatenate this to the end of the current list of values of k 
              Äż$ | Zip the cumulative sum of the values of k with the values

Jelly, 40 34 bytes

1SÆD>Ðḟ½ƊṀ;@Ʋ³¡żÄ$Ø.xṁ^þ`ʋ/€Ṛṛ""/Y

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1                  | Start with 1
            Ʋ³¡    | Loop through the following the number of times indicated by the first argument to the program; this generates a list of values of k
 S                 | - Sum
        Ɗ          | - Following three links as a monad 
  ÆD               |   - List of divisors
    >Ðḟ½           |   - Exclude those greater than the square root
         Ṁ         |   - Maximum
          ;@       | - Concatenate this to the end of the current list of values of k 
               żÄ$ | Zip the values of k with their cumulative sum

       ʋ/€ | For each pair of k and cumulative sum, call the following as a dyad with k as the left argument and the cumulative sum as the right
Ø.x        | - Repeat [0, 1] each k times (e.g. [0, 0, 0, 1, 1, 1] if k is 3
    ṁ       | - Mould  to the length of the cumulative sum
    ^þ`    | - Outer product using xor function and same argument to both side

Jelly, 40 33 bytes

1SÆD>Ðḟ½ƊṀ;@Ʋ³¡Äż$Ḷ:Ḃ^þ`ʋ/€Ṛṛ""/Y

Try it online!

1                  | Start with 1
            Ʋ³¡    | Loop through the following the number of times indicated by the first argument to the program; this generates a list of values of k
 S                 | - Sum
        Ɗ          | - Following three links as a monad 
  ÆD               |   - List of divisors
    >Ðḟ½           |   - Exclude those greater than the square root
         Ṁ         |   - Maximum
          ;@       | - Concatenate this to the end of the current list of values of k 
               Äż$ | Zip the cumulative sum of the values of k with the values

      ʋ/€ | For each pair of k and cumulative sum, call the following as a dyad with the cumulative sum of k as the left argument and k as the right (e.g. 15, 3)
Ḷ         | - Lowered range [0, 1 ... , 13, 14]
 :        | - Integer division by k [0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4]
  Ḃ       | - Mod 2 [0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0]
   ^þ`    | - Outer product using xor function and same argument to both side