How to run a cron job at every 5 hour interval - linux

Question

How to run a Cron job at every 5 hour interval?
I've tried this
0 */5 * * * /path/to/script/script.sh (Not necessarily a shell script)
But the problem with this expression is it'll run every 5 hour but the clock will reset at midnight 00:00:00 everyday.
Explanation: if the job executes at midnight 00:00:00 then following will be the execution timing
00:00:00, 05:00:00, 10:00:00, 15:00:00, 20:00:00
after 20:00:00 the job will execute at 00:00:00 which is 1 hour earlier than specified.
But my requirement is the job should run at 5 hour interval and shouldn't get reset at midnight
So if the job starts at midnight 00:00:00 the following should be the executing order.
00:00:00, 05:00:00, 10:00:00, 15:00:00, 20:00:00, (Day 1)
01:00:00, 06:00:00, 11:00:00, 16:00:00, 21:00:00, (Day 2)
02:00:00 ... (Day 3)
How do I achieve the following through Cron?
Any help would appreciated.

Answer 1

One option from this forum post on unix.com Run it every hour, and add to your script:
time_ct=$(cat /tmp/cron_time_ct)
if [ $time_ct -lt 5 ]
then
echo "Not yet time"
time_ct=$((time_ct+1))
echo $time_ct > /tmp/cron_time_ct
exit
else
time_ct=0
echo $time_ct > /tmp/cron_time_ct
fi
# rest of your task
I've done similar to do "every other monday" type logic. Running the script every monday and only allowing execution on every other one in the script.