Example 1
N,1,1,3,1,1,4,1
1---->2---->3->

Example 2
N,1,1,3,1,1,1,1
1---->2------->

Example
N,3,3,1,1,1,6,1
1>2>3------>4->

SELECT COUNT(*) + 1 FROM #temp1 WHERE DataDifference > 1

;With tblDifference as
(
Select Row_Number() OVER (Order by DateRecorded) as RowNumber,Name, DateRecorded from #m_TestTable
)
Select 
Cur.Name
,Cur.DateRecorded as CurrentDay
,Prv.DateRecorded as PreviousDay
,Datediff(Day, Prv.DateRecorded, Cur.DateRecorded) as DateDifference
,Cur.RowNumber
,0 Grp
into #temp1
From
tblDifference Cur Left Outer Join tblDifference Prv
On Cur.RowNumber = Prv.RowNumber + 1 
Order by Cur.DateRecorded

ALTER TABLE #temp1 ALTER COLUMN RowNumber int NOT NULL
ALTER TABLE #temp1 ADD CONSTRAINT tmp_pk PRIMARY KEY (RowNumber)

DECLARE @i AS INT = 1

UPDATE #temp1
SET Grp = @i,
	@i = @i + isnull(cast(cast(DateDifference - 1 as bit) as integer),0)

SELECT * FROM #temp1

WITH SampleData AS (
SELECT Name, RecordedDate
FROM (
Values ('John', dateadd(day,1,GetDate()))
,('John', dateadd(day,2,GetDate()))
,('John', dateadd(day,3,GetDate()))
,('John', dateadd(day,6,GetDate()))
,('John', dateadd(day,7,GetDate()))
,('John', dateadd(day,8,GetDate()))
, ('John', dateadd(day,12,GetDate()))
,('John', dateadd(day,13,GetDate())))a (Name, RecordedDate))
SELECT Name, COUNT(*)
FROM (
    SELECT Name
    FROM (
        SELECT Name, RecordedDate
            ,rn=RecordedDate-ROW_NUMBER() OVER (PARTITION BY Name ORDER BY RecordedDate)
        FROM SampleData) a
    GROUP BY Name, rn) b
GROUP BY Name