How to count the number of rows by datetime field with the division into time intervals

variable interval_minutes number;

exec :interval_minutes := 10;

select
  trunc(registered_at, 'HH24')
  + trunc(to_char(registered_at,'mi')/:interval_minutes)*:interval_minutes/1440, 
  count(*) 
from
  clients
group by 
  trunc(registered_at, 'HH24')
  + trunc(to_char(registered_at,'mi')/:interval_minutes)*:interval_minutes/1440
order by 
  1
;

Based on: Summarizing data over time - by time interval

SQL> select trunc(REGISTERED_AT,'hh')+trunc(to_char(REGISTERED_AT,'mi')/10)*10/1440 as "Start_Time",
      (trunc(REGISTERED_AT,'hh')+trunc(to_char(REGISTERED_AT,'mi')/10)*10/1440)+ (.000694 * 10) as "End_Time",
       count(*)
from clients
group by trunc(REGISTERED_AT,'hh')+trunc(to_char(REGISTERED_AT,'mi')/10)*10/1440,
         (trunc(REGISTERED_AT,'hh')+trunc(to_char(REGISTERED_AT,'mi')/10)*10/1440)+ (.000694 * 10);  2    3    4    5    6  

Start_Time          End_Time             COUNT(*)
------------------- ------------------- ----------
2016/08/01 15:00:00 2016/08/01 15:10:00      2
2016/08/01 16:30:00 2016/08/01 16:40:00      2
2016/08/01 15:40:00 2016/08/01 15:50:00      1

In case you need a more generic solution, you can use a function as this:

CREATE OR REPLACE FUNCTION MakeInterval(ts IN TIMESTAMP, roundInterval IN INTERVAL DAY TO SECOND) RETURN TIMESTAMP DETERMINISTIC IS
    denom INTEGER;
BEGIN
    IF roundInterval >= INTERVAL '1' HOUR THEN
        denom := EXTRACT(HOUR FROM roundInterval);
        IF MOD(24, denom) <> 0 THEN
            RAISE VALUE_ERROR;
        END IF;
        RETURN TRUNC(ts) + TRUNC(EXTRACT(HOUR FROM ts) / denom) * denom * INTERVAL '1' HOUR;
    ELSIF roundInterval >= INTERVAL '1' MINUTE THEN
        denom := EXTRACT(MINUTE FROM roundInterval);
        IF MOD(60, denom) <> 0 THEN
            RAISE VALUE_ERROR;
        END IF;
        RETURN TRUNC(ts, 'hh') + TRUNC(EXTRACT(MINUTE FROM ts) / denom) * denom * INTERVAL '1' MINUTE;
    ELSE
        denom := EXTRACT(SECOND FROM roundInterval);                
        IF MOD(60, denom) <> 0 THEN
            RAISE VALUE_ERROR;
        END IF;
        RETURN TRUNC(ts, 'mi') + TRUNC(EXTRACT(SECOND FROM ts) / denom) * denom * INTERVAL '1' SECOND;
    END IF;
END MakeInterval;



select count(*)
   MakeInterval(registered_at, INTERVAL '10' MINUTE) as start_,  
   MakeInterval(registered_at, INTERVAL '10' MINUTE) + INTERVAL '10' MINUTE as end_
from clients 
group by MakeInterval(registered_at, INTERVAL '10' MINUTE);

Please note, the function will raise an exception if the interval does not fit in whole number, for example INTERVAL '11' MINUTE would raise an error.