TITLE: "How many different integers does this give us?"
QUESTION [8 upvotes]: How many unique integers can you get from
$\lceil2012/n\rceil$
where $n$ is a positive integer?
I don't know at all where to begin to approach this problem. I thought it maybe had something to do with which factors the number has or anything like that, but I couldn't find any obvious pattern.

REPLY [2 votes]: In GAP it's three lines of code :
LoadPackage("numericalsgps");;
Set(List([1..2012],n->CeilingOfRational(2012/n)));
Size(last);

(albeit with one fairly redundant line of code).  This outputs:
[ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 
  22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 
  41, 42, 43, 44, 45, 46, 47, 48, 50, 51, 52, 53, 55, 56, 58, 60, 61, 63, 65, 
  68, 70, 72, 75, 78, 81, 84, 88, 92, 96, 101, 106, 112, 119, 126, 135, 144, 
  155, 168, 183, 202, 224, 252, 288, 336, 403, 503, 671, 1006, 2012 ]

and the number of results:
89