Consecutive dates count
You made me type your data, so this is as much as you get:

create table data
(
  claim_key varchar2(10 char), 
  mbr_id varchar2(10 char),
  serv_from date
);

insert into data values ('clm1', 'mem1', date'2018-01-01');
insert into data values ('clm2', 'mem1', date'2018-01-02');
insert into data values ('clm3', 'mem1', date'2018-01-03');
insert into data values ('clm3', 'mem1', date'2018-01-04');
insert into data values ('clm4', 'mem1', date'2018-01-07');
insert into data values ('clm5', 'mem1', date'2018-01-08');
insert into data values ('clm5', 'mem1', date'2018-01-09');
insert into data values ('clm5', 'mem1', date'2018-01-16');
commit;

select mbr_id, count(*)  from (
select mbr_id, sum(changed) over (partition by mbr_id order by serv_from) sum_changed from (
select mbr_id, serv_from,
  case when lag(serv_from) over (partition by mbr_id order by serv_from) - serv_from = -1 
    then 0 else 1 end changed 
from data
)) group by mbr_id, sum_changed;

MBR_ID       COUNT(*)
---------- ----------
mem1                4
mem1                3
mem1                1