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The built-in function 'input' appears to be sending the prompt to stderr instead of stdout, as the docs state it should.
$ python3.5 -c "print('stdout'); import sys; print('stderr', file=sys.stderr); input('foo')" 2> errors.txt
stdout
hello
$ cat errors.txt
stderr
foo
I've replicated this behavior in both Python 3.4 on Linux and Python 3.5 and 2.7 on OS X. Here's the 2.7 test/output:
$ python2.7 -c "from __future__ import print_function; print('stdout'); import sys; print('stderr', file=sys.stderr); raw_input('foo')" 2> errors.txt
stdout
hello
$ cat errors.txt 
stderr
foo
I believe the stated behavior (outputting the prompt to stdout) is the proper behavior.
I figure I must be doing something wrong here, because it seems bizarre to me that this discrepancy hasn't been discovered before.

This has been discovered before: Issue 1927, with a simple patch. The choice between stdout and stderr seems to be rather complicated, so the patch may need a bit more work.