Replacing multiple instances of a symbol with different variables and assignments

Question

I have an expression of the form:

expr=x+y+n+Sin[n]+n^2

I would like to replace each of the variables n with a different variable, like:

x+y+n1+Sin[n2]+n3^2

The order doesn't matter, but it matters that each n have a different name.

I can find the positions of the n's simply with:

pos=Position[expr,n]

and this returns a list of positions {{1}, {2, 1}, {5, 1}} as expected. I would then like to run through this list and make replacements in those positions. However, I don't want to have to do this by hand. Clearly I could do:

expr[[5,1]]=n3

but the real expression I have is many thousands of times larger.

I have tried using:

Apply[Part, Join[{expr}, pos[[3]]]]=n2

which would allow be to do this in a loop, or functionally, but this doesn't alter the element of the expression as I would like. I have tried playing with Hold in various places but it doesn't seem to work.

Anything simple that I am missing?

Answer 1

You may do something like:

expr = x + y + n + Sin[n] + n^2; 
{newExpr, {newVars}} = Reap[expr /. n :> Sow[Unique@n]];
{newExpr, newVars}
(*
 {n$9881 + n$9882^2 + x + y + Sin[n$9883], {n$9881, n$9882, n$9883}}
*)