I assume you use Sun-Sat week (as Aug 1st start on Sat, and you mention 6 weeks in Aug).
Cell to calculate # of partial weeks in a month.
If using Office 365.
Ex: In B1
=SUMPRODUCT(--(SEQUENCE(6,,A1-WEEKDAY(A1,2),7)<=EOMONTH(A1,0)))-(WEEKDAY(A1)=1)
Then in B3 to G3.
=IF(COLUMNS($A$1:A1)<=$B$1,"Week"&COLUMNS($A$1:A1),"")
Copy across.
If using older version of Excel/Office.
Change B1 formula to...
=SUMPRODUCT(--((A1-WEEKDAY(A1,2)+{0,7,14,21,28,35})<=EOMONTH(A1,0)))-(WEEKDAY(A1)=1)
See attached (J1 holds formula for older version).
EDIT: You can combine it into single formula. But for ease of following logic. It's separated into 2 parts.