#Perl, 78 bytes
#!perl -nl
grep$_<(($_+2)**(1/3)|0)**3,map$i**3-$_**3,2..$i++and$_-=print$i**3+1while$_
Brute force approach. Couting the shebang as two, input is taken from stdin.
Sample Usage
$ echo 10 | perl fermat-near-miss.pl
1729
1092728
3375001
15438250
121287376
401947273
3680797185
6352182209
7856862273
12422690497