Print the sequence

Scratch 3.0 39 blocks/323 bytes

Try it online scratch!

Alternatively, as SB syntax:

when gf clicked
delete[all v]of[o v
ask()and wait
set[. v]to(answer
set[n v]to(1
repeat until<(n)contains[.
if<((n)mod(4))=(0
set[. v]to((. )*(n
else
if<((n)mod(4))=(1
change[. v]by(n
else
if<((n)mod(4))=(2
change[. v]by((0)-(n
else
set[. v]to((. )/(n
end
end
end
add(n)to[o v
change[n v]by(1
end
delete(length of(o))of[o v

Look at you guys, having fun with your fancy eval statements! Well, not me! No... Scratch doesn't have evals, so I had to do things the hard way... if statements.

At least it isn't gotos...

Haskell, 75 74 73 bytes

-1 byte thanks to Will Ness -1 byte thanks to nimi

(#1)
n#i|i`mod`4<1,n`mod`i>0=[n]|y<-i+1=n:(x!!i)n i#y
x=div:(*):(+):(-):x

Try it online!

Avoids the use of fractional ints to save on bytes

Whitespace, 251 227 202 bytes

[S S S N
_Push_0][S N
S _Duplicate_0][S N
S _Duplicate_0][T   N
T   T   _Read_STDIN_as_integer][T   T   T   _Retrieve_input][N
S S N
_Create_Label_LOOP][S N
S _Duplicate_top][T N
S T _Print_as_integer][S S S T  S T S N
_Push_10_newline][T N
S S _Print_as_character][S N
T   _Swap_top_two][S S S T  N
_Push_1][T  S S S _Add][S T S S T   N
_Copy_2nd_item][S T S S T   N
_Copy_2nd_item][S N
S _Duplicate_top][S S S T   S S N
_Push_4][T  S T T   _Modulo][S N
S _Duplicate_top][N
T   S S N
_If_0_Jump_to_Label_DIVIDE][S S S T N
_Push_1][T  S S T   _Subtract][S N
S _Duplicate_top][N
T   S T N
_If_0_Jump_to_Label_MULTIPLY][S S S T   N
_Push_1][T  S S T   _Subtract][N
T   S S S N
_If_0_Jump_to_Label_ADD][S N
T   _Swap_top_two][S T  S S T   N
_Copy_2nd_item][T   S S T   _Subtract][N
S N
N
_Jump_to_LOOP][N
S S S N
_Create_Label_DIVIDE][S N
N
_Discard_top][T S T S _Divide][S T  S S T   S N
_Copy_3nd_item][S T S S T   S N
_Copy_3nd_item][T   S T T   _Modulo][N
T   S N
_If_0_Jump_to_Label_LOOP][N
N
N
_Exit_Program][N
S S T   N
_Create_Label_MULTIPLY][S N
N
_Discard_top][T S S N
_Multiply][N
S N
N
_Jump_to_Label_LOOP][N
S S S S N
_Create_Label_ADD][T    S S S _Add][N
S N
N
_Jump_to_Label_LOOP]

Letters S (space), T (tab), and N (new-line) added as highlighting only.
[..._some_action] added as explanation only.

Try it online (with raw spaces, tabs and new-lines only).

-24 bytes after a comment of @JoKing suggesting n%i > 0. Although only if(x < 0) and if(x == 0) are available in Whitespace, simply checking if(x*-1 < 0) is basically the same as if(x > 0).
An additional -25 bytes thanks to @JoKing.

Explanation:

Quote from the Scratch answer:

At least it isn't gotos...

Did someone say goto? Whitespace has nothing else than goto to create both loops and if-statements. xD In addition it's a stack-based language, so I have to swap/discard/copy pretty often. And to top things off: Whitespace doesn't even have floating points and only integer-division, so I've used n % i * -1 < 0 to exit the program if the integer cannot divide the current number.

Pseudo-code:

Integer n = STDIN as input
Integer i = 0
Label LOOP:
  Print n as number
  i = i + 1
  Integer t = i % 4
  If(t == 0):
    Jump to Label DIVIDE
  t = t - 1
  If(t == 0):
    Jump to Label MULTIPLY
  t = t - 1
  If(t == 0):
    Jump to Label ADD
  n = n - i
  Jump to Label LOOP
Label DIVIDE:
  n = n / i
  Integer m = n % i
  If(m == 0):
    Jump to Label LOOP
  Exit program
Label MULTIPLY:
  n = n * i
  Jump to Label LOOP
Label ADD:
  n = n + i
  Jump to Label LOOP