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I've found a few ways online to calculate the week number but none that allow me to start the week on Saturday. Here is the formula that I tested, but doesn't give me Saturday-Friday.

=INT(([Time In]-DATE(YEAR([Time In]),1,1)+(TEXT(WEEKDAY(DATE(YEAR([Time In]),1,1)),"d")))/7)+1
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If I understand you correctly, you could try below formula.

=ROUNDUP((Date1-DATE(YEAR(Date1),1,1)+1-(6-(TEXT(WEEKDAY(DATE(YEAR(Date1),1,1)),"d"))))/7,0)+1

Test Data:

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