Specifying "all odd values" in crontab?

In crontab, I can use an asterisk to mean every value, or "*/2" to mean every even value.

Is there a way to specify every odd value? (Would something like "1+*/2" work?)

Depending on your version of cron, you should be able to do (for hours, say):

   1-23/2

Going by the EXTENSIONS section in the crontab(5) manpage:

   Ranges can include "steps", so "1-9/2" is the same as "1,3,5,7,9".

For a more portable solution, I suspect you just have to use the simple list:

   1,3,5,7,9,11,13,15,17,19,21,23

But it might be easier to wrap your command in a shell script that will immediately exit if it's not called in an odd minute.

Every odd minute would be:

1-59/2 * * * * 

Every even minute would be:

0-58/2 * * * *